Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 896187 13 pageshttpdxdoiorg1011552013896187

Research ArticleAn Efficient Patch Dissemination Strategy for Mobile Networks

Dawei Zhao12 Haipeng Peng12 Lixiang Li12 Yixian Yang12 and Shudong Li3

1 Information Security Center State Key Laboratory of Networking and Switching TechnologyBeijing University of Posts and Telecommunications Beijing 100876 China

2National Engineering Laboratory for Disaster Backup and Recovery Beijing University of Posts and TelecommunicationsBeijing 100876 China

3 College of Mathematics and Information Science Shandong Institute of Business and Technology Shandong Yantai 264005 China

Correspondence should be addressed to Haipeng Peng penghaipengbupteducn

Received 8 June 2013 Accepted 5 July 2013

Academic Editor Ming Li

Copyright copy 2013 Dawei Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Mobile phones and personal digital assistants are becoming increasingly important in our daily life since they enable us to accessa large variety of ubiquitous services Mobile networks formed by the connection of mobile devices following some relationshipsamong mobile users provide good platforms for mobile virus spread Quick and efficient security patch dissemination strategy isnecessary for the update of antivirus software so that it can detect mobile virus especially the new virus under the wireless mobilenetwork environment with limited bandwidth which is also large scale decentralized dynamically evolving and of unknownnetwork topology In this paper we propose an efficient semi autonomy-oriented computing (SAOC) based patch disseminationstrategy to restrain the mobile virus In this strategy some entities are deployed in a mobile network to search for mobile devicesaccording to some specific rules and with the assistance of a center Through experiments involving both real-world networks anddynamically evolving networks we demonstrate that the proposed strategy can effectively send security patches to as many mobiledevices as possible at a considerable speed and lower cost in the mobile network It is a reasonable effective and secure method toreduce the damages mobile viruses may cause

1 Introduction

The last decade has witnessed a surge of wireless mobiledevices such as mobile phones PocketPCs netbooks andtablet PCs With the appearance and development of intel-ligent operating system mobile devices are getting smarterand more functional For example they can connect tothe Internet receive and send emails and short messages(SMS)multimedia messages (MMS) and connect to otherdevices for exchanging information and activating variousapplications Meanwhile these mobile devices also becomethe ideal targets of mobile virus because they are populardesigned to be open programmable and general of purposeand highly dependent on common software platforms such asAndroid Symbian Windows Mobile and Linux

Mobile networks formed by the connection of mobiledevices following some relationships among mobile usersprovide good platforms for mobile virus spread For

example an MMS-based worm named ldquoCommwarriorrdquo(httpwwwf-securecomv-descscommwarriorshtml) canspread in MMS network which is formed based on thesocial relationships among mobile users And a Bluetooth-based worm named ldquoCabirrdquo (httpwwwf-securecomv-descscabirshtml) can spread in Bluetooth network whichis formed according to the geographically positions of mobiledevices There have been extensive studies on modeling thevirusepidemic propagation [1ndash6] in complex networkswhich can be used to estimate the scale of a virusepidemicoutbreak before it actually occurs and evaluate the effectof new or improved countermeasures in restrainingvirusepidemic propagation And based on these studiesmany network immunization strategies [7ndash10] have beenproposed for restraining virus propagation by selectivelyimmunizing some nodes based on the measurements ofdegree or betweenness But it would be difficult for thesestrategies to deal with large-scale decentralized and dynamic

2 Mathematical Problems in Engineering

mobile networks Intrusion detection technology [11] isanother straight and effective means for the containment ofmobile virus However the detection capabilities of mostantivirus software are depend on the existence of an updatedvirus signature repository Antivirus users are not protectedwhenever an attacker spreads a previously never encounteredvirus In order to protect the mobile phones from the damageof new virus service providers or security companies needto quickly identify the new virus generate a signatureand disseminate patches to smart phones Currently mostresearches have been done on intrusion detection [11ndash13]and patch generation [14ndash16] while this paper aims to studythe dissemination [17ndash20] of security patch in the wirelessmobile network environment

Due to the limited bandwidth of wireless networks itis difficult to disseminate the security patches to all phonessimultaneously and timely And since the mobile networkis always large-scale decentralized dynamically and ofunknown network topology good patch dissemination strat-egy is necessary Some strategies attempt to forward securitynotifications or patches based on the short-range communi-cation capabilities of intermittently connected phones [17 18]These strategies select some important phones that can dividea Bluetooth-based network into different communities basedon the contact time and frequency Thereafter they sendsecurity signatures to all communities based on the localdetection However this method cannot ensure that usersacquire patches in time References [20 21] presented a quickand efficient autonomy-oriented computing (AOC) [22 23]based patch dissemination strategy based on SMS that can beused in multiple forms of mobile network But this strategystill has the following deficiencies (1) the number of patchesdisseminated is not determined at a time step Especiallythere may be many patches disseminated at the initial stagewhich can potentially cause network congestion [24 25] (2)a phone may receive the same patch from different neighborsmore than once which may lead to network congestion andthe waste of network resource Therefore it is still in highdemand to develop a new strategy that can efficiently andquickly send security patches to as many phones as possiblein the mobile network

In this paper we propose a patch dissemination strategybased on semi autonomy-oriented computing (SAOC) torestrain themobile virus For theAOC-based strategy certainentities reside in some phones in the mobile network Theyautonomously workwith each other andmove in the networkbased on their own autonomous behaviors But in our SAOC-based strategy a center is added to the AOC-based strategyto combine and analyze the information received from theentities At each time step each entity moves to the nextlocation according to its own autonomous behavior andthe information feedbacked from the center Through manyexperiments involving both synthetic and real-world net-works we find that the proposed SAOC-based strategy canquickly send security patches to as many phones as possiblein the mobile network with limited bandwidth which isalso large-scale decentralized dynamically and of unknownnetwork topology Besides it can control the number ofpatches disseminated at each time step and make adjustment

according to the network conditions The selected phoneswhich receive the patches are always the most importantones of the phones found by the entities at each time step forthe virus propagation and thus the virus propagation can beeffectively restrained The network congestion and the wasteof the network resources can also be avoided because eachphone receives the patch only once

2 SAOC-Based Patch Dissemination Strategy

SMSMMS messages and Bluetooth are becoming the twomajor propagation routes of mobile virus Since SMS-basedviruses are found more dangerous than Bluetooth-basedviruses in terms of propagation speed and scope [20]we propose a semi autonomy-oriented computing (SAOC)based patch dissemination strategy to restrain the SMS-based virus propagation in this paper For the autonomy-oriented computing (AOC) approach [20 26] a group ofcomputational entities are dispatched into a mobile networkThey reside in some phones autonomously work with eachothermove fromone phone to another andupdate their localenvironment based on their own autonomous behaviorsHowever in our SAOC-based approach the entities no longerwork full autonomously and a center is added to help theentities finish their tasks At each time step the center isresponsible for combining and analyzing the informationreceived from the entities and each entity moves from itspresent position to a new one according to some rules theinformation feedbacked from the center and the cooperationwith other entities We use a graph 119866 to denote the mobilephones network formed according to the address books ofmobile phones Some definitions which are used to formulatethe SAOC-based dissemination strategy are as follows

Definition 1 A graph 119866 = ⟨119881 119871⟩ is a mobile network formedaccording to the address books of mobile phones where 119881 =

V1 V2 V

119873 is a set of phones and 119871 = ⟨V

119894 V119895⟩|1 le 119894 119895 le

119873 119894 = 119895 is a set of undirected links (if V119894is in the address book

of V119895 then there is a link between V

119894and V119895 and V

119894is called a

friend of V119895)119873 = |119881| represents the total number of phones

in the networkEach phone V

119894in 119866 has two states ⟨119901ℎ119900119899119890119868119889

119886119897119897 119891119903119894119890119899119889119868119889119904⟩ where 119901ℎ119900119899119890119868119889 denotes the identifier ofV119894and 119891119903119894119890119899119889119868119889 is the identifier of the friend of V

119894

Definition 2 The center denoted by 119862 contains two states⟨119894119889 119905119886119904119896⟩ where 119894119889 denotes its identifier and 119905119886119904119896 stores aseries of its tasks

Definition 3 Let 119890 be an entity in a network 119866 Entity 119890

is represented by a tuple ⟨119894119889 119901ℎ119900119899119890119868119889 119886119897119897 119891119903119894119890119899119889119868119889119904

119897119894119891119890119888119910119888119897119890 119903119906119897119890⟩ where 119894119889 denotes the identifier of the entity119901ℎ119900119899119890119868119889 represents the identifier of the phone resided by119890 119891119903119894119890119899119889119868119889 is the identifier of the friend of the residedphone 119897119894119891119890119888119910119888119897119890 is the maximum time steps for an entityto reside on a phone and 119903119906119897119890 is a set which stores fourlocal behaviors of an entity including rational-moverational-jump random-jump and wait

Mathematical Problems in Engineering 3

V1

V2

V1

V2

V3 V3

V4

V6 V9 V6 V9

V7 V10V10

V8

V7

V8V11 V11

V12V12

V5

V4

V5

V1

V2

V3

V6 V9

V10V7

V8 V11

V12V4

V5

e1

e1

e1

e2e2

e2

Step 1 Step 2 Step 3

PhoneEntityPatched phoneResided phone

Local environment of entityHighest-remain degree phone in local environmentThe indirect interaction between two entitiesThe move behavior of entity

Figure 1 An example of the SAOC-based patch dissemination strategy

Definition 4 The local environment and prelocal informa-tion of an entity are denoted by119864

119897and119901119903119890119868

119897 respectively If an

entity 119890 resides onphone V119894 its local environment andprelocal

information are defined as 119864119897(119890) = V

119894 V119895 and 119901119903119890119868

119897(119890) =

V1015840119894119904 119894119889 amp V1015840

119894119904 119886119897119897 119891119903119894119890119899119889119868119889119904 V1015840

119895119904 119894119889 amp V1015840

119895119904 119886119897119897 119891119903119894119890119899119889119868119889119904

respectively where V119895 is the set of friends of V

119894

Definition 5 Remain degree of a phone denotes the numberof friends who have not received the patches of a phone Aphone is regarded as its own friend

At each time step each entity sends its prelocalinformation searched in its local environment to thecenter The center combines and analyzes the informationreceived from all entities according to its 119905119886119904119896 andshares the analysis results which are called the postlocalinformation 119901119900119904119905119868

119897(119890) with each entity 119890 where 119901119900119904119905119868

119897(119890) =

V1015840119894119904 119894119889amp V1015840

119894119904 119903119890119898119886119894119899 119889119890119892119903119890119890 V1015840

119895119904 119894119889amp V1015840

119895119904 119903119890119898119886119894119899 119889119890119892119903119890119890

V119895 is the set of friends of V

119894resided by 119890 If two phones

resided by two entities are friends or they have at least a samefriend we assume that these two entities can share theirpostlocal information Each entity then moves to the nextlocation (119905119886119903119892119890119905119868119889) according to its 119903119906119897119890 Algorithm 1 showsthe detailed process of SAOC-based patch disseminationstrategy

The 119905119886119904119896 of the center includes the following

(1) Delete the 119891119903119894119890119899119889119868119889119904 who have received the patchesfrom each phonersquos 119886119897119897 119891119903119894119890119899119889119868119889119904 in all the prelocalinformation

(2) Compute each phonersquos remain degree and send thesecurity patches to the first 119898 phones with thehighest-remain degree (Therefore the number ofpatches disseminated at each time step is controllablethat can be adjusted according to the network con-ditions) And record the 119894119889119904 of the phones who justreceived the patches

(3) Delete the new patched 119891119903119894119890119899119889119868119889119904 from each phonersquos119886119897119897 119891119903119894119890119899119889119868119889119904 and compute each phonersquos new remaindegree

(4) Send the postlocal information to the entity

The main behaviors of each entity are as follows

(1) Rational move An entity moves to a phone with thehighest-remain degree in its postlocal information orthe shared postlocal information if it exists If thereexists more than one highest-remain degree phonethe entity will randomly choose one for residing in

(2) Rational jump the entity requests from the center aphone for residing in if such phone exists

(3) Random jump an entity moves along the edges witha randomly-determined number of steps in order toavoid getting stuck in local optima

(4) Wait If an entity does not find any available phone forresiding in it will stay at its current position

For example as shown in Figure 1 two entities 1198901and 1198902

reside in phones V5and V6at the initial phase of step 1 respec-

tively 1198901and 1198902begin to search their local environments and

obtain the prelocal information as

119901119903119890119868119897(1198901) = V

5amp V5 V4 V4amp V4 V1 V2 V5 V7 V8

119901119903119890119868119897(1198902) = V

6amp V6 V7 V9 V10

V7amp V7 V4 V6 V8 V10 V9amp V9 V6 V10 V12

V10

amp V10 V6 V7 V8 V9 V11 V12

(1)

When receiving 119901119903119890119868119897(1198901) and 119901119903119890119868

119897(1198902) the center firstly

deletes the phonesrsquo 119894119889 that has been immunized fromeach phonesrsquo 119891119903119894119890119899119889119868119889119904 and computes the remain degreeof each phone Since there are no phones have beenimmunized the remain degree of each phone will beV4amp 6 V

5amp 2 V

6amp 4 V

7amp 5 V

9amp 4 V

10amp 7 In this

4 Mathematical Problems in Engineering

0 100 200 300

0

200

400

600

800

1000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

H = 50

(a) University email network

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500

4000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

H = 50

(b) Community-based network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

H = 50

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

H = 50

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 2 The number of infected phones over time

moment the center sends the security patches to the first5 unimmunized phones (in this example we assume thatno more than 119898 = 5 phones are immunized at each timestep) with highest-remain degree that is V

4 V6 V7 V9 V10

and deletes these phonesrsquo 119894119889 from each phonesrsquo 119891119903119894119890119899119889119868119889119904and computes the new remain degree of each phone Thenew remain degree will be sent to entities as their postlocalinformation that is 119901119900119904119905119868

119897(1198901) = V

5amp 1 V

4amp 4 and

119901119900119904119905119868119897(1198902) = V

6amp 0 V

7amp 1 V

9amp 1 V

10amp 3 When

receiving the postlocal information each entity will moveto the phone which has the highest-remain degree in itspostlocal information Therefore 119890

1and 119890

2move from V

5

to V4and from V

6to V10 respectively In this step these

two entities perform the rational move relying on theirown postlocal information Step 2 will show the case of themovement of the entities relying on the shared postlocalinformation In step 2 when 119890

1and 1198902receive 119901119900119904119905119868

119897(1198901) and

119901119900119904119905119868119897(1198902) from the center they can share their postlocal

Mathematical Problems in Engineering 5

0 50 100 150 200 2500

200

400

600

800

1000

1200

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(a) University email network

Time step0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

3500

4000

4500

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Time step0 100 200 300 400

0

2000

4000

6000

8000

10000

12000

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

12000

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 3 The number of immunized phones over time

information with each other since they have the mutualfriends V

7and V

8 119901119900119904119905119868

119897(1198901) 119901119900119904119905119868

119897(1198902) and the shared

postlocal information are as follows

119901119900119904119905119868119897(1198901)=V4amp 1 V

1amp 0 V2amp 1 V5amp 1 V7amp 0 V8amp 0

119901119900119904119905119868119897(1198902) = V

10amp 0

V6amp 0 V

7amp 0 V

8amp 0

V9amp 0 V

11amp 0 V

12amp 0

119901119900119904119905119868119897(1198901) cup 119901119900119904119905119868

119897(1198902)

= V1amp 0 V

2amp 1 V

4amp 1 V

5amp 1 V

6amp 0

V7amp 0 V

8amp 0 V

9amp 0 V

10amp 0 V

11amp 0 V

12amp 0

(2)

1198901and 119890

2will choose the first two phones with the

highest-remain degree in the shared postlocal informationas their target locations Note that there are three phonescan be resided and 119890

1is residing in one of the highest-

remain degree phone In this case we let 1198901continue from

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

mobile networks Intrusion detection technology [11] isanother straight and effective means for the containment ofmobile virus However the detection capabilities of mostantivirus software are depend on the existence of an updatedvirus signature repository Antivirus users are not protectedwhenever an attacker spreads a previously never encounteredvirus In order to protect the mobile phones from the damageof new virus service providers or security companies needto quickly identify the new virus generate a signatureand disseminate patches to smart phones Currently mostresearches have been done on intrusion detection [11ndash13]and patch generation [14ndash16] while this paper aims to studythe dissemination [17ndash20] of security patch in the wirelessmobile network environment

Due to the limited bandwidth of wireless networks itis difficult to disseminate the security patches to all phonessimultaneously and timely And since the mobile networkis always large-scale decentralized dynamically and ofunknown network topology good patch dissemination strat-egy is necessary Some strategies attempt to forward securitynotifications or patches based on the short-range communi-cation capabilities of intermittently connected phones [17 18]These strategies select some important phones that can dividea Bluetooth-based network into different communities basedon the contact time and frequency Thereafter they sendsecurity signatures to all communities based on the localdetection However this method cannot ensure that usersacquire patches in time References [20 21] presented a quickand efficient autonomy-oriented computing (AOC) [22 23]based patch dissemination strategy based on SMS that can beused in multiple forms of mobile network But this strategystill has the following deficiencies (1) the number of patchesdisseminated is not determined at a time step Especiallythere may be many patches disseminated at the initial stagewhich can potentially cause network congestion [24 25] (2)a phone may receive the same patch from different neighborsmore than once which may lead to network congestion andthe waste of network resource Therefore it is still in highdemand to develop a new strategy that can efficiently andquickly send security patches to as many phones as possiblein the mobile network

In this paper we propose a patch dissemination strategybased on semi autonomy-oriented computing (SAOC) torestrain themobile virus For theAOC-based strategy certainentities reside in some phones in the mobile network Theyautonomously workwith each other andmove in the networkbased on their own autonomous behaviors But in our SAOC-based strategy a center is added to the AOC-based strategyto combine and analyze the information received from theentities At each time step each entity moves to the nextlocation according to its own autonomous behavior andthe information feedbacked from the center Through manyexperiments involving both synthetic and real-world net-works we find that the proposed SAOC-based strategy canquickly send security patches to as many phones as possiblein the mobile network with limited bandwidth which isalso large-scale decentralized dynamically and of unknownnetwork topology Besides it can control the number ofpatches disseminated at each time step and make adjustment

according to the network conditions The selected phoneswhich receive the patches are always the most importantones of the phones found by the entities at each time step forthe virus propagation and thus the virus propagation can beeffectively restrained The network congestion and the wasteof the network resources can also be avoided because eachphone receives the patch only once

2 SAOC-Based Patch Dissemination Strategy

SMSMMS messages and Bluetooth are becoming the twomajor propagation routes of mobile virus Since SMS-basedviruses are found more dangerous than Bluetooth-basedviruses in terms of propagation speed and scope [20]we propose a semi autonomy-oriented computing (SAOC)based patch dissemination strategy to restrain the SMS-based virus propagation in this paper For the autonomy-oriented computing (AOC) approach [20 26] a group ofcomputational entities are dispatched into a mobile networkThey reside in some phones autonomously work with eachothermove fromone phone to another andupdate their localenvironment based on their own autonomous behaviorsHowever in our SAOC-based approach the entities no longerwork full autonomously and a center is added to help theentities finish their tasks At each time step the center isresponsible for combining and analyzing the informationreceived from the entities and each entity moves from itspresent position to a new one according to some rules theinformation feedbacked from the center and the cooperationwith other entities We use a graph 119866 to denote the mobilephones network formed according to the address books ofmobile phones Some definitions which are used to formulatethe SAOC-based dissemination strategy are as follows

Definition 1 A graph 119866 = ⟨119881 119871⟩ is a mobile network formedaccording to the address books of mobile phones where 119881 =

V1 V2 V

119873 is a set of phones and 119871 = ⟨V

119894 V119895⟩|1 le 119894 119895 le

119873 119894 = 119895 is a set of undirected links (if V119894is in the address book

of V119895 then there is a link between V

119894and V119895 and V

119894is called a

friend of V119895)119873 = |119881| represents the total number of phones

in the networkEach phone V

119894in 119866 has two states ⟨119901ℎ119900119899119890119868119889

119886119897119897 119891119903119894119890119899119889119868119889119904⟩ where 119901ℎ119900119899119890119868119889 denotes the identifier ofV119894and 119891119903119894119890119899119889119868119889 is the identifier of the friend of V

119894

Definition 2 The center denoted by 119862 contains two states⟨119894119889 119905119886119904119896⟩ where 119894119889 denotes its identifier and 119905119886119904119896 stores aseries of its tasks

Definition 3 Let 119890 be an entity in a network 119866 Entity 119890

is represented by a tuple ⟨119894119889 119901ℎ119900119899119890119868119889 119886119897119897 119891119903119894119890119899119889119868119889119904

119897119894119891119890119888119910119888119897119890 119903119906119897119890⟩ where 119894119889 denotes the identifier of the entity119901ℎ119900119899119890119868119889 represents the identifier of the phone resided by119890 119891119903119894119890119899119889119868119889 is the identifier of the friend of the residedphone 119897119894119891119890119888119910119888119897119890 is the maximum time steps for an entityto reside on a phone and 119903119906119897119890 is a set which stores fourlocal behaviors of an entity including rational-moverational-jump random-jump and wait

Mathematical Problems in Engineering 3

V1

V2

V1

V2

V3 V3

V4

V6 V9 V6 V9

V7 V10V10

V8

V7

V8V11 V11

V12V12

V5

V4

V5

V1

V2

V3

V6 V9

V10V7

V8 V11

V12V4

V5

e1

e1

e1

e2e2

e2

Step 1 Step 2 Step 3

PhoneEntityPatched phoneResided phone

Local environment of entityHighest-remain degree phone in local environmentThe indirect interaction between two entitiesThe move behavior of entity

Figure 1 An example of the SAOC-based patch dissemination strategy

Definition 4 The local environment and prelocal informa-tion of an entity are denoted by119864

119897and119901119903119890119868

119897 respectively If an

entity 119890 resides onphone V119894 its local environment andprelocal

information are defined as 119864119897(119890) = V

119894 V119895 and 119901119903119890119868

119897(119890) =

V1015840119894119904 119894119889 amp V1015840

119894119904 119886119897119897 119891119903119894119890119899119889119868119889119904 V1015840

119895119904 119894119889 amp V1015840

119895119904 119886119897119897 119891119903119894119890119899119889119868119889119904

respectively where V119895 is the set of friends of V

119894

Definition 5 Remain degree of a phone denotes the numberof friends who have not received the patches of a phone Aphone is regarded as its own friend

At each time step each entity sends its prelocalinformation searched in its local environment to thecenter The center combines and analyzes the informationreceived from all entities according to its 119905119886119904119896 andshares the analysis results which are called the postlocalinformation 119901119900119904119905119868

119897(119890) with each entity 119890 where 119901119900119904119905119868

119897(119890) =

V1015840119894119904 119894119889amp V1015840

119894119904 119903119890119898119886119894119899 119889119890119892119903119890119890 V1015840

119895119904 119894119889amp V1015840

119895119904 119903119890119898119886119894119899 119889119890119892119903119890119890

V119895 is the set of friends of V

119894resided by 119890 If two phones

resided by two entities are friends or they have at least a samefriend we assume that these two entities can share theirpostlocal information Each entity then moves to the nextlocation (119905119886119903119892119890119905119868119889) according to its 119903119906119897119890 Algorithm 1 showsthe detailed process of SAOC-based patch disseminationstrategy

The 119905119886119904119896 of the center includes the following

(1) Delete the 119891119903119894119890119899119889119868119889119904 who have received the patchesfrom each phonersquos 119886119897119897 119891119903119894119890119899119889119868119889119904 in all the prelocalinformation

(2) Compute each phonersquos remain degree and send thesecurity patches to the first 119898 phones with thehighest-remain degree (Therefore the number ofpatches disseminated at each time step is controllablethat can be adjusted according to the network con-ditions) And record the 119894119889119904 of the phones who justreceived the patches

(3) Delete the new patched 119891119903119894119890119899119889119868119889119904 from each phonersquos119886119897119897 119891119903119894119890119899119889119868119889119904 and compute each phonersquos new remaindegree

(4) Send the postlocal information to the entity

The main behaviors of each entity are as follows

(1) Rational move An entity moves to a phone with thehighest-remain degree in its postlocal information orthe shared postlocal information if it exists If thereexists more than one highest-remain degree phonethe entity will randomly choose one for residing in

(2) Rational jump the entity requests from the center aphone for residing in if such phone exists

(3) Random jump an entity moves along the edges witha randomly-determined number of steps in order toavoid getting stuck in local optima

(4) Wait If an entity does not find any available phone forresiding in it will stay at its current position

For example as shown in Figure 1 two entities 1198901and 1198902

reside in phones V5and V6at the initial phase of step 1 respec-

tively 1198901and 1198902begin to search their local environments and

obtain the prelocal information as

119901119903119890119868119897(1198901) = V

5amp V5 V4 V4amp V4 V1 V2 V5 V7 V8

119901119903119890119868119897(1198902) = V

6amp V6 V7 V9 V10

V7amp V7 V4 V6 V8 V10 V9amp V9 V6 V10 V12

V10

amp V10 V6 V7 V8 V9 V11 V12

(1)

When receiving 119901119903119890119868119897(1198901) and 119901119903119890119868

119897(1198902) the center firstly

deletes the phonesrsquo 119894119889 that has been immunized fromeach phonesrsquo 119891119903119894119890119899119889119868119889119904 and computes the remain degreeof each phone Since there are no phones have beenimmunized the remain degree of each phone will beV4amp 6 V

5amp 2 V

6amp 4 V

7amp 5 V

9amp 4 V

10amp 7 In this

4 Mathematical Problems in Engineering

0 100 200 300

0

200

400

600

800

1000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

H = 50

(a) University email network

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500

4000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

H = 50

(b) Community-based network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

H = 50

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

H = 50

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 2 The number of infected phones over time

moment the center sends the security patches to the first5 unimmunized phones (in this example we assume thatno more than 119898 = 5 phones are immunized at each timestep) with highest-remain degree that is V

4 V6 V7 V9 V10

and deletes these phonesrsquo 119894119889 from each phonesrsquo 119891119903119894119890119899119889119868119889119904and computes the new remain degree of each phone Thenew remain degree will be sent to entities as their postlocalinformation that is 119901119900119904119905119868

119897(1198901) = V

5amp 1 V

4amp 4 and

119901119900119904119905119868119897(1198902) = V

6amp 0 V

7amp 1 V

9amp 1 V

10amp 3 When

receiving the postlocal information each entity will moveto the phone which has the highest-remain degree in itspostlocal information Therefore 119890

1and 119890

2move from V

5

to V4and from V

6to V10 respectively In this step these

two entities perform the rational move relying on theirown postlocal information Step 2 will show the case of themovement of the entities relying on the shared postlocalinformation In step 2 when 119890

1and 1198902receive 119901119900119904119905119868

119897(1198901) and

119901119900119904119905119868119897(1198902) from the center they can share their postlocal

Mathematical Problems in Engineering 5

0 50 100 150 200 2500

200

400

600

800

1000

1200

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(a) University email network

Time step0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

3500

4000

4500

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Time step0 100 200 300 400

0

2000

4000

6000

8000

10000

12000

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

12000

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 3 The number of immunized phones over time

information with each other since they have the mutualfriends V

7and V

8 119901119900119904119905119868

119897(1198901) 119901119900119904119905119868

119897(1198902) and the shared

postlocal information are as follows

119901119900119904119905119868119897(1198901)=V4amp 1 V

1amp 0 V2amp 1 V5amp 1 V7amp 0 V8amp 0

119901119900119904119905119868119897(1198902) = V

10amp 0

V6amp 0 V

7amp 0 V

8amp 0

V9amp 0 V

11amp 0 V

12amp 0

119901119900119904119905119868119897(1198901) cup 119901119900119904119905119868

119897(1198902)

= V1amp 0 V

2amp 1 V

4amp 1 V

5amp 1 V

6amp 0

V7amp 0 V

8amp 0 V

9amp 0 V

10amp 0 V

11amp 0 V

12amp 0

(2)

1198901and 119890

2will choose the first two phones with the

highest-remain degree in the shared postlocal informationas their target locations Note that there are three phonescan be resided and 119890

1is residing in one of the highest-

remain degree phone In this case we let 1198901continue from

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

V1

V2

V1

V2

V3 V3

V4

V6 V9 V6 V9

V7 V10V10

V8

V7

V8V11 V11

V12V12

V5

V4

V5

V1

V2

V3

V6 V9

V10V7

V8 V11

V12V4

V5

e1

e1

e1

e2e2

e2

Step 1 Step 2 Step 3

PhoneEntityPatched phoneResided phone

Local environment of entityHighest-remain degree phone in local environmentThe indirect interaction between two entitiesThe move behavior of entity

Figure 1 An example of the SAOC-based patch dissemination strategy

Definition 4 The local environment and prelocal informa-tion of an entity are denoted by119864

119897and119901119903119890119868

119897 respectively If an

entity 119890 resides onphone V119894 its local environment andprelocal

information are defined as 119864119897(119890) = V

119894 V119895 and 119901119903119890119868

119897(119890) =

V1015840119894119904 119894119889 amp V1015840

119894119904 119886119897119897 119891119903119894119890119899119889119868119889119904 V1015840

119895119904 119894119889 amp V1015840

119895119904 119886119897119897 119891119903119894119890119899119889119868119889119904

respectively where V119895 is the set of friends of V

119894

Definition 5 Remain degree of a phone denotes the numberof friends who have not received the patches of a phone Aphone is regarded as its own friend

At each time step each entity sends its prelocalinformation searched in its local environment to thecenter The center combines and analyzes the informationreceived from all entities according to its 119905119886119904119896 andshares the analysis results which are called the postlocalinformation 119901119900119904119905119868

119897(119890) with each entity 119890 where 119901119900119904119905119868

119897(119890) =

V1015840119894119904 119894119889amp V1015840

119894119904 119903119890119898119886119894119899 119889119890119892119903119890119890 V1015840

119895119904 119894119889amp V1015840

119895119904 119903119890119898119886119894119899 119889119890119892119903119890119890

V119895 is the set of friends of V

119894resided by 119890 If two phones

resided by two entities are friends or they have at least a samefriend we assume that these two entities can share theirpostlocal information Each entity then moves to the nextlocation (119905119886119903119892119890119905119868119889) according to its 119903119906119897119890 Algorithm 1 showsthe detailed process of SAOC-based patch disseminationstrategy

The 119905119886119904119896 of the center includes the following

(1) Delete the 119891119903119894119890119899119889119868119889119904 who have received the patchesfrom each phonersquos 119886119897119897 119891119903119894119890119899119889119868119889119904 in all the prelocalinformation

(2) Compute each phonersquos remain degree and send thesecurity patches to the first 119898 phones with thehighest-remain degree (Therefore the number ofpatches disseminated at each time step is controllablethat can be adjusted according to the network con-ditions) And record the 119894119889119904 of the phones who justreceived the patches

(3) Delete the new patched 119891119903119894119890119899119889119868119889119904 from each phonersquos119886119897119897 119891119903119894119890119899119889119868119889119904 and compute each phonersquos new remaindegree

(4) Send the postlocal information to the entity

The main behaviors of each entity are as follows

(1) Rational move An entity moves to a phone with thehighest-remain degree in its postlocal information orthe shared postlocal information if it exists If thereexists more than one highest-remain degree phonethe entity will randomly choose one for residing in

(2) Rational jump the entity requests from the center aphone for residing in if such phone exists

(3) Random jump an entity moves along the edges witha randomly-determined number of steps in order toavoid getting stuck in local optima

(4) Wait If an entity does not find any available phone forresiding in it will stay at its current position

For example as shown in Figure 1 two entities 1198901and 1198902

reside in phones V5and V6at the initial phase of step 1 respec-

tively 1198901and 1198902begin to search their local environments and

obtain the prelocal information as

119901119903119890119868119897(1198901) = V

5amp V5 V4 V4amp V4 V1 V2 V5 V7 V8

119901119903119890119868119897(1198902) = V

6amp V6 V7 V9 V10

V7amp V7 V4 V6 V8 V10 V9amp V9 V6 V10 V12

V10

amp V10 V6 V7 V8 V9 V11 V12

(1)

When receiving 119901119903119890119868119897(1198901) and 119901119903119890119868

119897(1198902) the center firstly

deletes the phonesrsquo 119894119889 that has been immunized fromeach phonesrsquo 119891119903119894119890119899119889119868119889119904 and computes the remain degreeof each phone Since there are no phones have beenimmunized the remain degree of each phone will beV4amp 6 V

5amp 2 V

6amp 4 V

7amp 5 V

9amp 4 V

10amp 7 In this

4 Mathematical Problems in Engineering

0 100 200 300

0

200

400

600

800

1000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

H = 50

(a) University email network

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500

4000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

H = 50

(b) Community-based network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

H = 50

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

H = 50

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 2 The number of infected phones over time

moment the center sends the security patches to the first5 unimmunized phones (in this example we assume thatno more than 119898 = 5 phones are immunized at each timestep) with highest-remain degree that is V

4 V6 V7 V9 V10

and deletes these phonesrsquo 119894119889 from each phonesrsquo 119891119903119894119890119899119889119868119889119904and computes the new remain degree of each phone Thenew remain degree will be sent to entities as their postlocalinformation that is 119901119900119904119905119868

119897(1198901) = V

5amp 1 V

4amp 4 and

119901119900119904119905119868119897(1198902) = V

6amp 0 V

7amp 1 V

9amp 1 V

10amp 3 When

receiving the postlocal information each entity will moveto the phone which has the highest-remain degree in itspostlocal information Therefore 119890

1and 119890

2move from V

5

to V4and from V

6to V10 respectively In this step these

two entities perform the rational move relying on theirown postlocal information Step 2 will show the case of themovement of the entities relying on the shared postlocalinformation In step 2 when 119890

1and 1198902receive 119901119900119904119905119868

119897(1198901) and

119901119900119904119905119868119897(1198902) from the center they can share their postlocal

Mathematical Problems in Engineering 5

0 50 100 150 200 2500

200

400

600

800

1000

1200

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(a) University email network

Time step0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

3500

4000

4500

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Time step0 100 200 300 400

0

2000

4000

6000

8000

10000

12000

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

12000

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 3 The number of immunized phones over time

information with each other since they have the mutualfriends V

7and V

8 119901119900119904119905119868

119897(1198901) 119901119900119904119905119868

119897(1198902) and the shared

postlocal information are as follows

119901119900119904119905119868119897(1198901)=V4amp 1 V

1amp 0 V2amp 1 V5amp 1 V7amp 0 V8amp 0

119901119900119904119905119868119897(1198902) = V

10amp 0

V6amp 0 V

7amp 0 V

8amp 0

V9amp 0 V

11amp 0 V

12amp 0

119901119900119904119905119868119897(1198901) cup 119901119900119904119905119868

119897(1198902)

= V1amp 0 V

2amp 1 V

4amp 1 V

5amp 1 V

6amp 0

V7amp 0 V

8amp 0 V

9amp 0 V

10amp 0 V

11amp 0 V

12amp 0

(2)

1198901and 119890

2will choose the first two phones with the

highest-remain degree in the shared postlocal informationas their target locations Note that there are three phonescan be resided and 119890

1is residing in one of the highest-

remain degree phone In this case we let 1198901continue from

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0 100 200 300

0

200

400

600

800

1000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

H = 50

(a) University email network

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500

4000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

H = 50

(b) Community-based network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

H = 50

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

Time step

The n

umbe

r of i

nfec

ted

phon

es

H = 50

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 2 The number of infected phones over time

moment the center sends the security patches to the first5 unimmunized phones (in this example we assume thatno more than 119898 = 5 phones are immunized at each timestep) with highest-remain degree that is V

4 V6 V7 V9 V10

and deletes these phonesrsquo 119894119889 from each phonesrsquo 119891119903119894119890119899119889119868119889119904and computes the new remain degree of each phone Thenew remain degree will be sent to entities as their postlocalinformation that is 119901119900119904119905119868

119897(1198901) = V

5amp 1 V

4amp 4 and

119901119900119904119905119868119897(1198902) = V

6amp 0 V

7amp 1 V

9amp 1 V

10amp 3 When

receiving the postlocal information each entity will moveto the phone which has the highest-remain degree in itspostlocal information Therefore 119890

1and 119890

2move from V

5

to V4and from V

6to V10 respectively In this step these

two entities perform the rational move relying on theirown postlocal information Step 2 will show the case of themovement of the entities relying on the shared postlocalinformation In step 2 when 119890

1and 1198902receive 119901119900119904119905119868

119897(1198901) and

119901119900119904119905119868119897(1198902) from the center they can share their postlocal

Mathematical Problems in Engineering 5

0 50 100 150 200 2500

200

400

600

800

1000

1200

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(a) University email network

Time step0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

3500

4000

4500

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Time step0 100 200 300 400

0

2000

4000

6000

8000

10000

12000

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

12000

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 3 The number of immunized phones over time

information with each other since they have the mutualfriends V

7and V

8 119901119900119904119905119868

119897(1198901) 119901119900119904119905119868

119897(1198902) and the shared

postlocal information are as follows

119901119900119904119905119868119897(1198901)=V4amp 1 V

1amp 0 V2amp 1 V5amp 1 V7amp 0 V8amp 0

119901119900119904119905119868119897(1198902) = V

10amp 0

V6amp 0 V

7amp 0 V

8amp 0

V9amp 0 V

11amp 0 V

12amp 0

119901119900119904119905119868119897(1198901) cup 119901119900119904119905119868

119897(1198902)

= V1amp 0 V

2amp 1 V

4amp 1 V

5amp 1 V

6amp 0

V7amp 0 V

8amp 0 V

9amp 0 V

10amp 0 V

11amp 0 V

12amp 0

(2)

1198901and 119890

2will choose the first two phones with the

highest-remain degree in the shared postlocal informationas their target locations Note that there are three phonescan be resided and 119890

1is residing in one of the highest-

remain degree phone In this case we let 1198901continue from

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 50 100 150 200 2500

200

400

600

800

1000

1200

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(a) University email network

Time step0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

3500

4000

4500

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Time step0 100 200 300 400

0

2000

4000

6000

8000

10000

12000

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0 100 200 300 400 5000

2000

4000

6000

8000

10000

12000

Time step

The n

umbe

r of i

mm

uniz

ed p

hone

s

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 3 The number of immunized phones over time

information with each other since they have the mutualfriends V

7and V

8 119901119900119904119905119868

119897(1198901) 119901119900119904119905119868

119897(1198902) and the shared

postlocal information are as follows

119901119900119904119905119868119897(1198901)=V4amp 1 V

1amp 0 V2amp 1 V5amp 1 V7amp 0 V8amp 0

119901119900119904119905119868119897(1198902) = V

10amp 0

V6amp 0 V

7amp 0 V

8amp 0

V9amp 0 V

11amp 0 V

12amp 0

119901119900119904119905119868119897(1198901) cup 119901119900119904119905119868

119897(1198902)

= V1amp 0 V

2amp 1 V

4amp 1 V

5amp 1 V

6amp 0

V7amp 0 V

8amp 0 V

9amp 0 V

10amp 0 V

11amp 0 V

12amp 0

(2)

1198901and 119890

2will choose the first two phones with the

highest-remain degree in the shared postlocal informationas their target locations Note that there are three phonescan be resided and 119890

1is residing in one of the highest-

remain degree phone In this case we let 1198901continue from

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

Time step0 50 100 150 200 250 300 350

0

200

400

600

800

1000

1200

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

0 50 100 150 200 250 300 3500

2000

4000

6000

8000

10000

Time step

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Time step0 50 100 150 200 250 300 350

0

1000

2000

3000

4000

5000

The n

umbe

r of p

atch

es

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

(d) Collaboration network

Figure 4 The number of patches over time

moving Therefore 1198901and 119890

2move from V

4to V5and

V10

to V2 respectively Table 1 presents the detailed patch

dissemination process of Figure 1 based on our SAOC-basedpatch dissemination strategy

3 Experimentation and Validation

31 Static Networks A mobile network is constructedbased on the address books of smart phones whichreflects the social relationship among mobile users in realworld situations Here we use some benchmark networks

(university email network autonomous systemsnetwork andcollaboration network) to reflect the relationship structuresin the real world Table 2 shows the structure and degree offour networks University email network [27] autonomoussystems network [28] and collaboration network of ArxivHigh Energy Physics category [29] are real-world networksCommunity-based network is a synthetic network with fourcommunities based on the GLP algorithm [30]

We use the four networks shown in Table 2 to evaluate theefficiency of the proposed SAOC-based patch disseminationstrategy in restraining the SMS-based virus For the SMS-based virus propagation model we assume the following

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

20

40

60

80

100

Coverage

The a

vera

ge o

f eac

h en

tity

steps

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0 02 04 06 08 10

50

100

150

200

250

300

Coverage

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

(b) Community-based network

Coverage0 02 04 06 08 1

0

50

100

150

200

250

300

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

Coverage0 02 04 06 08 1

0

100

200

300

400

500

The a

vera

ge o

f eac

h en

tity

steps

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 5 The average of each entity steps with respect to coverage rate

(1) If a user receives a message from his friend hemay open or delete this message determined by hissecurity awareness [20 31 32]The security awarenessof different users in this paper is consistent withthat used by [20] and follows a normal distribution119873(05 03

2)

(2) If a user opens a virus message he is infected and willautomatically send the virusmessage to all his friends

(3) An infected phone sends the virus to his friends onlyonce after which the infected phone will not send outvirus any more

(4) If a phone has received the patch it will not send outvirus even if the user opens an infectedmessage again

At some point we deploy a few entities into a mobilenetwork These entities reside in the phones with the highestdegree which are found by the AOC-based immuniza-tion strategy [26] Each entity then moves according to

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

5000

10000

15000

20000

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 50

SAOC entity = 10 m = 50

SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

(a) University email network

0

05

1

15

2

25

3

Coverage rate

The t

otal

num

ber o

f pat

ches

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 100

SAOC entity = 10 m = 100

SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

times104

(b) Community-based network

0

1

2

3

4

5

Coverage rate

The t

otal

num

ber o

f pat

ches

times104

0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 300

SAOC entity = 10 m = 300

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(c) Autonomous systems network

0

05

1

15

2

25

The t

otal

num

ber o

f pat

ches

times105

Coverage rate0 02 04 06 08 1

AOC entity = 5AOC entity = 10SAOC entity = 5 m = 200

SAOC entity = 10 m = 200

SAOC entity = 5 m = 500

SAOC entity = 10 m = 500

(d) Collaboration network

Figure 6 The number of patches with respect to coverage rate

Algorithm 1 We compare the efficiency of our SAOC-baseddissemination strategy with the AOC-based disseminationstrategy [20] by different indexes in the above static bench-mark networks

Figure 2 shows the average numbers of infected phonesover time when 5 and 10 entities are deployed into thenetworks from the time step of 50 At each time step nomore than119898 patches can be sent in SAOC-based strategy thatis up to 119898 phones can be immunized at each time step in

SAOC-based strategy Obviously the earlier and themore thepatch is disseminated the shorter the propagation durationwill be Figure 3 shows the average number of immunizedphones over time when the entities are deployed into thenetworks from50 Sincewe set a limit on the size of119898 to avoidthe network congestion the effect of SAOC-based strategyis inferior to the AOC-based one at the initial phase afterthe deploying of the entities when119898 is small But simulationresults show that the SAOC-based strategy can recover all the

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(1) For each entity 119890search the local environment 119864

119897(119890) and obtain pre-local information 119901119903119890119868

119897(119890)

send 119901119903119890119868119897(119890) to the center

End(2) For center 119862

perform a series of task according to its taskEnd

(3) For each entity 119890compute targetId based on the post-local information or the shared post-local informationIf targetId is not nullThen

Rational move to targetIdElse if elifecycle lt 1Then

request the center a targetIdIf receive a targetIdThen

Rational jump to targetIdElse if not receive a targetIdThen

Random jump to targetIdElse

WaitEnd

End

Algorithm 1 The process of SAOC-based patch dissemination strategy

infected phones and immune all the phones faster than theAOC-based one even if 119898 is relatively small Figure 4 showsthe number of the patches disseminated at each time stepWefind that the number of the patches disseminated at each timestep in AOC-based strategy is much more than that of theSAOC-based one Figure 4 also shows the main inadequaciesof AOC-based strategy that is too many patches are sent atcertain times which may lead to network congestion and aphone may receive the patch from different neighbors morethan once which causes the waste of network resourcesHowever in our SAOC-based strategy the number of patchesdisseminated at each time step is controllable that can beadjusted according to the network conditions and a phonereceives the patch only once

Figures 5 and 6 show the average number of steps ofeach entity and the total number of patches disseminatedcorresponding to the coverage rate respectivelyThe coveragerate is defined as 119873immunized119873 where 119873immunized representsthe total number of immunized phones that are patched bythe center and 119873 represents the total number of phones inthe network In Figure 5 each entity in SAOC-based strategyneeds to move a bit more steps than that in the AOC-basedstrategy when the coverage rate is small due to the limitationon 119898 But in the case of achieving a significant amount ofcoverage rate the number of steps of each entity needed tomove is much smaller in SAOC-based strategy than that inAOC-based strategy In Figure 6 we can see that the totalnumber of patches disseminated is much smaller in SAOC-based strategy than in AOC-based strategy to attain the samecoverage rate

From the simulations performed above we can see thatthe SAOC-based dissemination strategy can efficiently sendsecurity patches to as many phones as possible with consid-erable speed and relatively lower cost in the static networks

32 Dynamically Evolving Networks In this section weevaluate the efficiency of SAOC-based dissemination strategyin dynamically evolving networks since the structure of anetwork is changing in the real world We assume that theinitial network contains 1000 phones with ⟨119896⟩ = 8 Threedifferent patterns of network evolving are considered asfollows (1) the network scale will grow to 4000 (2) 50 or100 phones are added into the network at each step fromthe time step of 20 (3) the network degree ⟨119896⟩ will remainunchanged or change from 8 to 18 respectively We usethe SIR [33ndash35] model to characterize the SMS-based viruspropagation in dynamically evolving networks SIR is themost basic and well-studied epidemic spreading model Inthe SIR model the elements of a network are divided intothree compartments including susceptibles (S those whocan contract the infection) infectious (I those who havecontracted the infection and are contagious) and recovered(R those who have recovered from the disease) At each timestep we assume that a susceptible phone becomes infectedwith a probability 120582 if it is directly connected to an infectedphone Meanwhile if an infected phone receives the patch itwill become to be recovered from the infected state

Simulation results shown in Figure 7 indicate that whenselecting the appropriate number of patches disseminated ateach time step our SAOC-based strategy can send securitypatches to as many phones as possible and reduce the dam-ages of mobile virus in the dynamically evolving networkswith various complex evolving patterns

4 Conclusion

In this paper we propose an efficient SAOC-based patchdissemination strategy to restrain the SMS-based mobile

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 The detailed process of Figure 1 based on SAOC-based patch dissemination strategy

Entity Center Entity

Step 1Entity

Analytical information

Step 2

Store immune phones

Store immune phones

Step 3

Store immune phones

Entity

Entity Entity

Entity Entity

Entity Entity

Entity

Entity Entity

Pre-local information Post-local information Movement

then

their post-local informations

end

e1 and e2 share

e1V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V9 amp V9 V6 V10 V12

V10 amp V10 V6 V7 V8 V9 V11 V12

e1V4 amp V4 V1 V2 V5 V7 V8

V1 amp V2 V1 V4

V2 amp V1 V2 V3 V4

V5 amp V5 V4

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

e2V10 amp V10 V6 V7 V8 V9 V11 V12

V6 amp V6 V7 V9 V10

V7 amp V7 V4 V6 V8 V10

V8 amp V8 V4 V7 V10 V11

V9 amp V9 V6 V10 V12

V11 amp V11 V8 V10 V12

V12 amp V12 V9 V10 V11

e1

V2 amp V2 V1 V3 V4

V1 amp V2 V1 V4

V3 amp V3 V2

V4 amp V4 V1 V2 V5 V7 V8

V5 amp V5 V4

V4 amp V4 V1 V2 V5 V7 V8

e2

e1

Entity e2

e1

e2

e1

e2

V4 amp V4 V1 V2 V5 V7 V8 amp 6 amp 4

V4 amp V4 V1 V2 V5 V7 V8 amp 4 amp 1

V5 amp V5 V4 amp 2 amp 1

V6 amp V6 V7 V9 V10 amp 4 amp 0

V2 amp V2 V1 V3 V4 amp 3 amp 1

V7 amp V7 V4 V6 V8 V10 amp 5 amp 1

V9 amp V9 V6 V10 V12 amp 4 amp 1

V10 amp 3

V10 amp 0

V11 amp 0

V12 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 7 amp 3

V4 V6 V7 V9 V10

V4 amp 4

V6 amp 0

V7 amp 1

V9 amp 1

V4 amp 1

V1 amp 0

V1 amp 0

V2 amp 0

V3 amp 0

V4 amp 0

V4 amp 0

V5 amp 0

V2 amp 1

V5 amp 1

V7 amp 0

V8 amp 0

V8 amp 0

V9 amp 0

V6 amp 0

V7 amp 0

V5 amp 1

V1 amp V1 V2 V4 amp 2 amp 0

V5 amp V5 V4 amp 1 amp 1

V6 amp V6 V7 V9 V10 amp 0 amp 0

V7 amp V7 V4 V6 V8 V10 amp 1 amp 0

V8 amp V8 V4 V7 V10 V11 amp 2 amp 0

V9 amp V9 V6 V10 V12 amp 1 amp 0

V10 amp V10 V6 V7 V8 V9 V11 V12 amp 3 amp 0

V11 amp V11 V8 V10 V12 amp 3 amp 0

V12 amp V12 V9 V10 V11 amp 2 amp 0

V1 amp V2 V1 V4 amp 0 amp 0

V2 amp V2 V1 V3 V4 amp 1 amp 0

V3 amp V3 V2 amp 1 amp 0

V4 amp V4 V1 V2 V5 V7 V8 amp 1 amp 0

V5 amp V5 V4 amp 1 amp 0

V1 V2 V4 V6 V7 V8 V9 V10 V11 V12

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

e1 V5 rarr V4

e1 V4 rarr V5

e2 V6 rarr V10

e2 V10 rarr V2

In the analytical information V119894 amp V1198951 V119895119896 amp 1198991 amp 1198992 of the center V119894 is the identifier of a phone V1198951 V119895119896 the friends of V119894 1198991 the first computedremain degree of V119894 and 1198992 the second computed remain degree of V119894 The identifiers in red indicate the phones that have received the patches in theprevious steps The identifiers in blue indicate the phones that will receive the patches in the current step The no more than 5 red numbers in each steprefers to the unimmunized phones with the highest-first computed remain degree

virus The advantages of our SAOC-based strategy could bedescribed as follows

(1) it sends security patches to asmany phones as possibleat a considerable speed and lower cost in the mobile

network with limited bandwidth which is also large-scale decentralized dynamically evolving and ofunknown network topology

(2) it can control the number of patches disseminated ateach time step and make adjustment according to the

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 8 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(a)

0 100 200 300 400 500

0

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 50 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(b)

0 100 200 300 400 5000

500

1000

1500

2000

The n

umbe

r of i

nfec

ted

phon

es

Time step

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

k = 8 to 8 each step 100 nodes

(c)

0 100 200 300 400 5000

500

1000

1500

2000

Time step

The n

umbe

r of i

nfec

ted

phon

es

k = 8 to 18 each step 100 nodes

AOC entity = 3AOC entity = 6CAOC entity = 3 m = 50

CAOC entity = 3 m = 100

CAOC entity = 6 m = 50

CAOC entity = 6 m = 100

(d)

Figure 7 The number of infected phones over time in different dynamicin-evolving networks (a) 50 phones are added into the network ateach step and the average degree ⟨119896⟩maintains 8 (b) 50 phones are added into the network at each step and the average degree ⟨119896⟩ increasesfrom 8 to 18 (c) 100 phones are added into the network at each step and the average degree ⟨119896⟩ maintains 8 (d) 100 phones are added intothe network at each step and the average degree ⟨119896⟩ increases from 8 to 18

network conditionsThus the network congestion canbe avoided

(3) the selected phones which receive the patches arealways the most important ones of the phones foundby the entities at each time step for the virus propaga-tion and thus the virus propagation can be effectivelyrestrained

(4) each phone receives the patch only once which isbeneficial to avoiding the network congestion and thewaste of network resource

In summary the SAOC-based patch dissemination strat-egy is a reasonable effective and secure method to sendsecurity patches in mobile networks and reduce the damagesmobile viruses cause

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 2 The structures of networks

Nodes Edges ⟨119896⟩

University email network 1133 5451 962Community-based network 4000 16855 842Autonomous systems network 11080 31538 569Collaboration network 12008 237010 3947

Acknowledgments

This paper was supported by the National Natural Sci-ence Foundation of China (Grant nos 61202362 6107020961121061 and 61272402) the Asia Foresight Program underNSFC Grant (Grant no 61161140320) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20120005110017)

References

[1] P Wang M C Gonzalez C A Hidalgo and A-L BarabasildquoUnderstanding the spreading patterns of mobile phonevirusesrdquo Science vol 324 no 5930 pp 1071ndash1076 2009

[2] S-M Cheng W C Ao P-Y Chen and K-C Chen ldquoOnmodelingmalware propagation in generalized social networksrdquoIEEE Communications Letters vol 15 no 1 pp 25ndash27 2011

[3] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth worms (Extended version)rdquo IEEE Transactions onMobile Computing vol 8 no 3 pp 353ndash367 2009

[4] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[5] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[6] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[7] Y Chen G Paul S Havlin F Liljeros and H E Stanley ldquoFind-ing a better immunization strategyrdquo Physical Review Letters vol101 no 5 Article ID 058701 2008

[8] W-J Bai T Zhou and B-H Wang ldquoImmunization ofsusceptible-infected model on scale-free networksrdquo Physica Avol 384 no 2 pp 656ndash662 2007

[9] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 4 pages 2003

[10] P Holme B J Kim C N Yoon and S K Han ldquoAttackvulnerability of complex networksrdquo Physical Review E vol 65no 5 Article ID 056109 14 pages 2002

[11] D Samfat and R Molva ldquoIDAMN an intrusion detectionarchitecture for mobile networksrdquo IEEE Journal on SelectedAreas in Communications vol 15 no 7 pp 1373ndash1380 1997

[12] T S Yap and H T Ewe ldquoA mobile phone malicious softwaredetection model with behavior checkerrdquo in Web and Commu-nication Technologies and Internet-Related Social IssuesmdashHSI2005 vol 3597 of Lecture Notes in Computer Science pp 57ndash652005

[13] J Cheng S H Y Wong H Yang and S Lu ldquoSmartSiren Virusdetection and alert for smartphonesrdquo in Proceedings of the 5thInternational Conference on Mobile Systems Applications andServices (MobiSys rsquo07) pp 258ndash271 June 2007

[14] A Smirnov and T-C Chiueh ldquoAutomatic patch generation forbuffer overflow attacksrdquo in Proceedings of the 3rd InternationlSymposium on Information Assurance and Security (IAS rsquo07) pp165ndash170 August 2007

[15] S Sidiroglou andA D Keromytis ldquoCountering network wormsthrough automatic patch generationrdquo IEEE Security and Pri-vacy vol 3 no 6 pp 41ndash49 2005

[16] W Cui M Peinado H J Wang andM E Locasto ldquoShieldGenautomatic data patch generation for unknown vulnerabilitieswith informed probingrdquo in Proceedings of the IEEE Symposiumon Security and Privacy (SP rsquo07) pp 252ndash266 May 2007

[17] F Li Y Yang and JWu ldquoCPMC an efficient proximitymalwarecoping scheme in smartphone-based mobile networksrdquo in Pro-ceedings of the IEEE Conference on Computer Communications(INFOCOM rsquo10) pp 1ndash9 March 2010

[18] G Zyba G M Voelker M Liljenstam A Mehes and PJohansson ldquoDefending mobile phones from proximity mal-warerdquo in Proceedings of the 28th Conference on ComputerCommunications (INFOCOM rsquo09) pp 1503ndash1511 April 2009

[19] M H R Khouzani S Sarkar and E Altman ldquoDispatch thenstop optimal dissemination of security patches in mobilewireless networksrdquo in Proceedings of the 49th IEEE Conferenceon Decision and Control (CDC rsquo10) pp 2354ndash2359 December2010

[20] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing vol 12no 3 pp 529ndash541 2013

[21] C Gao and J Liu ldquoModeling and restraining mobile viruspropagationrdquo IEEE Transactions on Mobile Computing 2013

[22] J Liu ldquoAutonomy-Oriented Computing (AOC) the natureand implications of a paradigm for self-organized computingrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 3ndash11 October 2008

[23] J Liu X Jin and K C Tsui Autonomy Oriented Computing(AOC) From Problem Solving to Complex Systems ModelingSpringer New York NY USA 2005

[24] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 154038 11 pages 2012

[25] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[26] C Gao J Liu and N Zhong ldquoNetwork immunization withdistributed autonomy-oriented entitiesrdquo IEEE Transactions onParallel and Distributed Systems vol 22 no 7 pp 1222ndash12292011

[27] RGuimera LDanonADıaz-Guilera F Giralt andAArenasldquoSelf-similar community structure in a network of humaninteractionsrdquoPhysical Review E vol 68 no 6 Article ID 065103pp 651031ndash651034 2003

[28] J Leskovec J Kleinberg and C Faloutsos ldquoGraphs over timedensification laws shrinking diameters and possible explana-tionsrdquo in Proceedings of the 11th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining pp 177ndash187 ACM August 2005

[29] J Leskovec J Kleinberg and C Faloutsos ldquoGraph evolutiondensification and shrinking diametersrdquo ACM Transactions onKnowledge Discovery from Data vol 1 no 1 pp 1ndash41 2007

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[30] T Bu and D Towsley ldquoOn distinguishing between internetpower law topology generatorsrdquo in Proceedings of the 21st IEEEInternational Conference on Computer and Communications(INFOCOM rsquo02) pp 638ndash647 2002

[31] C C Zou D F Towsley and W Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 106ndash118 2007

[32] C Gao J Liu andN Zhong ldquoNetwork immunization and viruspropagation in email networks experimental evaluation andanalysisrdquo Knowledge and Information Systems vol 27 no 2 pp253ndash279 2011

[33] T J Norman Bailey The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London UK 2nd edition1975

[34] D J Daley and J Gani Epidemic Modelling An IntroductionCambridge University Press Cambridge UK 2000

[35] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons New York NYUSA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of