Modeling Opportunistic Social Networks with Decayed ... · Modeling Opportunistic Social Networks with Decayed Aggregation Graph . Peiyan Yuan1, 2 and Yali Wang1. 1School of Computer

Modeling Opportunistic Social Networks with Decayed

Aggregation Graph

Peiyan Yuan1, 2

and Yali Wang1

1School of Computer and Information Engineering, Henan Normal University, Xinxiang, China

2Engineering Laboratory of Intellectual Business and Internet of Things Technologies, Henan Province

Email: [email protected], [email protected]

Abstract—The emerging mobile applications including

contact-based forwarding and online worm containment have

put a heavy burden on opportunistic social networks (OSNs),

the first goal is to develop an effective model that can reveal the

hidden, important social features underlying the OSNs. This

problem is especially challenging because of the time-varying

network topology. Traditional time expanded graph caches each

snapshot of networks, resulting in low computation efficiency

and high storage overhead. By aggregating past contact events

between two nodes to a simple boolean indicator, the binary

graph model can alleviate this issue. However, it only provides

a coarse-grained level of identifying the relationship between

nodes. It neglects the differences in contact events. Intuitively,

recent contact events are generally more important than old

ones, and in computing an aggregation, we should assign bigger

weights to them. In addition, since each contact has its own

duration, we should take this factor into account as well.

Motivated by these observations, we propose DAG, a decayed

aggregation graph for modeling OSNs at a fine-grained level.

By implementing DAG in different real scenarios, we show that

DAG is efficient in characterizing the relationship between

nodes and in improving the performance of mobile applications.

We simultaneously prove that DAG achieves approximate space

complexity as the binary graph.

Index Terms—Opportunistic social networks, decayed

aggregation graph, worm diffusion, data forwarding

I. INTRODUCTION

The pervasive deployment of portable devices (such as

smart phones) has resulted in a demand for opportunistic

social networks to support emerging mobile applications

such as contact-based forwarding and online worm

containment [1]. However, the success of these mobile

applications is restrained by the lack of effective model

that can reveal the hidden, important social relationships

underlying the OSNs [2]. Indeed, detecting the interesting

social features is of considerable advantage in mobile

applications. Consider the contact-based forwarding

problem in OSNs [3]. Due to the intermittently connected

links between nodes, an apparent challenge for any

Manuscript received January 1, 2015; revised March 27, 2015. This work was supported by the National Natural Science

Foundation of China under Grant No. U1404602, the Science and

Technology Foundation of Henan Educational Committee under Grant No.14A520031 and the Dr. Startup Project of Henan Normal University

under Grant No. qd14136. Corresponding author email: [email protected].

forwarding algorithm is to deliver the packet as quickly

as possible, without introducing many replicas of packets.

Since people have different social relationships (e.g.,

friend, acquaintance or stranger [4], there are also

different similarity metrics in the underlying network as a

reflection. A desired forwarding method is therefore to

select relays having high similarity scores with the

destination, instead of purely forwarding replicas to each

encountered node.

Another great advantage of exploiting social features

can be found in the online worm containment [5]-[6].

Suppose that a popular mobile application is infected

with malicious codes, the first thing is to prevent worms

from spreading massively by immunizing central nodes.

This example, again, highlights the big role of social

features in mobile applications. Unfortunately, it is very

challenging to identify them in OSNs, this is mainly

because of the time-varying network topology.

An ideal solution to the above problems would try to

duplicate each snapshot of the network as the traditional

time expanded graph [7]-[8] or its variant [9] did.

Nonetheless, this method suffers from some major

disadvantages: a) the huge storage overhead due to the

replication of the network, and b) the low computation

efficiency because of the increased problem size. To this

end, the authors of [10]-[13] tend to aggregate node

contacts by using a binary graph, where a link forms if

two nodes have at least a contact in the past time window

W. The binary graph achieves a lightweight overhead, but

it can only answer simple queries such as “Did Alice

meet up Bob in the past W ?”. In other words, it identifies

node relationship at a coarse-grained level, in the form of

aggregating past contact events between two nodes to a

boolean indicator. It obviously neglects the differences in

contact events. Intuitively, recent contact events are

generally more important than old ones, and in computing

an aggregate, we should assign bigger weights to them. In

addition, since each contact event has its own duration,

we should consider this factor as well.

Motivated by the observations, we propose DAG, a

decayed aggregation graph to model OSNs at a fine-

grained level. For any two nodes, we take the duration of

each contact and its decayed age into account, and

formulate their relationship strength as a decayed sum

problem (a decayed age a is equal to iT e , where T is the

current time and ei is the end moment of the ith contact,

respectively). We summarize our contributions as follows:

213

Journal of Communications Vol. 10, No. 3, March 2015

doi:10.12720/jcm.10.3.213-220

©2015 Journal of Communications

We propose DAG, a decayed aggregation graph for

not only tracing the relationship between nodes over

time but also revealing the social features underlying

the OSNs at a fine-grained level.

We prove that DAG achieves approximate space

complexity compared to the binary graph.

We validate DAG on different real opportunistic

scenarios. The experimental results show that DAG is

effective in characterizing the relationship between

nodes and in improving the performance of mobile

applications.

The remainder of this paper is organized as follows.

Section II briefly reviews the related work. Section III

presents the motivation. Section IV gives our solution.

We have a comparative analysis with binary graph in

Section V, followed by a short conclusion in Section VI.

(a) GTW (t=3000s) (b) STW (t=3000s, W=3000s)

(c) DAG (t=3000s, λ=1) (d) GTW (t=6000s)

(e) STW (t=6000s, W=3000s) (f) DAG (t=6000s, λ=1)

Fig. 1. Adjacency matrixes of three models at different time instances. Black → 0 and white → 1. We here take the data-set KAIST as a sample. For more information of the real data-sets, please refer to the section V.

II. RELATED WORK

Modeling the time-varying network topology has

attracted much attention since its introduction. Based on

the underlying principles, we generally classify the

existing models in two categories including time

expanded graph and the binary graph.

Time expanded graph [7]-[9] can accurately identify

the social relationship between nodes by caching each

snapshot of the original network. New edges connecting a

node and its copy are added at the next snapshot in

addition to the edges having been existed in the last

snapshot. Therefore, the network size increases quickly

with the elapsed time. Furthermore, the method suffers

from the low computation efficiency as well, because of

the increased problem size. To this end, recent works

[10]-[13] tend to aggregate node contacts by using a time

window. For example, the authors of [10] employed

growing time window to compute the social features over

a social graph, where an edge emerges if there exists at

least one contact between two nodes at any time in the

past. A sliding time window is used in [11] and [12],

where time is split into a sequence of time frames with

same size W and only contacts in the last time frame form

edges of graph. In addition, metrics such as most recent

contacts and most frequent contacts were taken in [13].

214



Compared to the time expanded graph, the proposed

binary graph indeed alleviates the storage overhead.

However, this method neglects the duration of each

contact and its decayed age. It only characterizes the

time-varying network at a coarse-grained level. Detailed

comparison between DAG and the binary graph is

presented in section IV.

III. MOTIVATION

We take the binary graph with growing time window

(GTW) and that with sliding time window (STW) as two

examples to analyze the possible problem caused by the

traditional solutions. Let N denote number of nodes in the

network. Let uv N NR r t

and uv N N

S s t

denote the

adjacency matrixes of GTW and STW, respectively. We

have 1uvr if node u and node v have at least a contact at

any time in the past, otherwise 0uvr , and 1uvs if they

have at least a contact in the past time window W,

otherwise 0uvs .

Apparently, the matrixes will get more and more

identical with the contacts aggregation as shown in Fig. 1

(a), Fig. 1 (d), Fig. 1 (b) and Fig. 1 (e). As a consequence,

heterogeneity of nodes cannot be well reflected, which in

turn degenerates the networking performance (please

refer to the section V). Hence, we conclude that the

binary graph only characterizes the OSNs at a coarse

level. On the contrary, if we model OSNs by a decayed

aggregation graph (please refer to the next section), we

achieve a fine-grained level of characterizing relationship

between nodes (Fig. 1(c) and Fig. 1 (f)).

IV. DECAYED GRAPH

A. DAG

We model an opportunistic social network as a

Decayed Aggregation Graph (DAG) G=(V, E), where V

denotes set of nodes in the network and E denotes set of

edges [14]. Let uvh t denote the relationship strength

between nodes u and v and ( ) uv N NH t h

denote G’s

adjacency matrix at moment t. We discuss how to

compute uvh t by aggregating their past contact events

as follows.

Fig. 2. Decayed aggregation graph and its components.

Let n denote the number of contact events between two

nodes (in general, n N ). We consider the contact series

1 2, ,..., nC C C C . Each element iC is a tuple ,i is e as

shown in the upper subfigure of Fig. 2, where ,i is e and

id denotes the start time, end time and duration of the ith

contact, respectively. As discussed above, the aggregate

will be time-decayed. That is, older contact events will be

decayed faster than more recent ones. The exact model of

decay is in general specified by a user through a decayed

function [15] [16].

Definition 1(Decayed Function): We call a function

0f x x , a decayed function, if it satisfies (a)

0 ( ) 1f x and (b) if 1 2x x , then 1 2f x f x .

At current time T, the decay of ei is defined as iT e .

At the same time, the intensity of each contact event

should be considered as well, since each contact has

different durations. The differentiated intensity is denoted

by a weighted function.

Definition 2 (Weighted Function): A function

0w x x is called a weighted function, if it holds (a)

0w x and (b) if 1 2x x , then 1 2w x w x .

For each contact iC , its weight is defined as

i ie s .

Problem 1(Decayed Sum Problem): Given the contact

series C, functions f x and w x , the first goal is to

estimate the decayed sum at current time T.

1

n

i ii

uvh T w x f x

(1)

where xi denotes the ith contact, i i iw x e s and

i if x f T e .

Since the inter-contact time between nodes always

follows an exponential decay [17], we further set

iT e

if T e e

. Hence, the Eq. (1) can be

reformulated as

( )

1

i

nT e

uv i i

i

h T e s e

(2)

The bottom subfigure of Fig. 2 shows the decayed

aggregation graph and its components, where the left

subfigure presents the weight function i iw e s and

decayed function if t e (the shadow part, ie t T )

of the ith contact, and the right part denotes the value of

uvh t . We next analyze the space complexity of DAG.

B. Space Complexity of DAG

Obviously, exact tracking of uvh t needs ( )n

storage bits. Considering scalability issue, we should

further reduce the storage overhead while keeping the

same calculation precision.

Let i ig t e s if t equals to ei, otherwise, 0g t ,

we obtain the following lemma 1.

Lemma 1: In a continuous interval [0,T], the Eq. (2) is

equivalent to the following Eq. (3)

215


GGREGATIONA


( )T t

uv t Th T g t e

(3)

Proof: Let us split the interval [0, T] into two disjoined

parts D1 and D2, where `

1

( [ , ])n

i i i i

i

D d d s e

and 1 2 [0, ]D D T .

We have

1 2

( ) ( ) ( )( ) ( ) ( )T t T t T t

t T t D t D

g t e g t e g t e

For any 2t D , since

it e (note that 1ie D and

1 2D D ), we have g(t)=0. Hence,

1

( ) ( )

( )

1

( )

1

( ) ( )

( )

( )

= ( )

i

i

T t T t

t T t D

nT t

i t d

nT e

i ii

uv

g t e g t e

g t e

e sd e

h T

Theorem 1: At each time slot 0,1,2,...,s T , the value

of ( )uvh T can be maintained easily using

( ) ( ) ( 1)uv uvh T g T e h T (4)

Proof: From the Lemma 1, we know

( )

( ) ( )

1

( 1 1)

1

( 1 )

1

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( 1)

T s

uv

s T

T T T s

s T

T s

s T

T s

s T

uv

h T g s e

g T e g s e

g T g s e

g T e g s e

g T e h T

Theorem 2: The space complexity of DAG is 2(2 )O N .

Proof: From the Theorem 1, we know that each node

only requires carrying one counter to exactly track the

relationship between it and any other nodes. Since there

are total N nodes, we need ( ( 1))O N N storage bits, plus

the 2( )O N storage bits required by the adjacency

matrix H , we get the conclusion.

TABLE I: SPACE COMPLEXITY OF TEG, GTW, STW AND DAG

Four models TEG[8] GTW STW DAG

Space

complexity 2( )O nN 2( )O N 2(2 )O N 2(2 )O N

C. Discussion of the Space Complexity

Table I lists the space complexity of time expanded

graph (TEG), GTW, STW and DAG. We can see that

DAG has the same space complexity as that of STW, and

both of them have a slightly higher space complexity than

GTW, as the latter does not need to store additional

contact information. Note that for DAG, we can achieve

the approximate space complexity even we exploit other

popular decayed functions such as polynomial decay and

poly-exponential decay. Please refer to [15] for more

information about the impact of decayed function on

storage overhead.

By taking two important social features, centrality and

similarity of nodes as samples, we next observe which of

the DAG, GTW and STW can best extract such

knowledge from the underlying OSNs. In addition, we

also evaluate the impact of similarity metrics revealed by

the three models on opportunistic forwarding, and that of

centrality on worm diffusion speed, two interesting

mobile applications.

V. PERFORMANCE EVALUATION

A. Data-Sets

We use two real data-sets, referred to as North

Carolina State Fair and KAIST. Both of them have been

applied into different scenarios (e.g., message deletion

mechanism in [18] and the localization of mobile

networks in [19]).

In KAIST, 34 students who live in a campus dormitory

carried the GPS devices (GPS 60CSx) from 2006-09-26

to 2007-10-03 and altogether 92 daily traces were

gathered. In Statefair, 19 traces were gathered from 18

volunteers who visited a local state fair that includes

many street arcades, small street food stands and

showcases. Each participant in this site spent less than

three hours. More than one thousand people daily were

attracted for two weeks due to the popular event. Besides,

the site is completely outdoor. The detailed statistics of

the two data-sets can be found in [20].

B. Evaluating Centrality

We first explore the impact of the three models on

evaluating centrality metric. Node centrality reflects the

relative importance of a node in the network. There exist

a lot of methods to measure the centrality. In this paper,

we employ the betweenness measure [21], one of the

most popular methods [22], to evaluate centrality.

Betweenness centrality reflects the controlling capability

of a node to other nodes, which measures the extent to

which a node falls on the shortest paths between two

other nodes. The higher the betweenness centrality of a

node is, the bigger the ability it has to facilitate

communication to other nodes within the network is.

Betweenness centrality of a node i is computed as:

1 1

( )N Nuv

ij k uv

l iB

l

(5)

where luv is the total number of shortest path between

nodes u and v, and luv(i) is the number of those paths that

include node i .

Currently, the exact calculation of Eq. (5) is to flood a

large number of messages in the network, and count the

number of times a node acts as a relay for other nodes on

all the shortest paths. We call this method Flooding and

take it as a benchmark. We next approximately estimate

Bi by simulating the message propagation process [23].

216



For a concise presentation, let uvx denote the contact

relationship between nodes u and v measured by the

above three models (i.e., uv uvx h in DAG, uv uvx r in

GTW or uv uvx s in STW). For any randomly generated

message m with source node s and destination node d,

when node s encounters node v, node v is selected as a

relay with a normalized probability

1

( )

( )v N

i

vP

i

(6)

where the term ( )i denote the degree of node i.

(a) Flooding versus DAG

(b) Flooding versus GTW

(c) Flooding versus STW

Fig. 3. Fast Fourier Transform (FFT) of centrality distribution at KAIST.

Node v takes the same infection strategy if it is

selected. The above process is repeated until the message

m reaches the destination node d. Thus, a path from s to d

is collected. Using this method, we can observe which

node plays a big role in the messages propagation process

by computing its appearance frequency in the shortest

paths we collect.

We present our results in Fig. 3 and Fig. 4, where the

centrality distributions of nodes are presented by the Fast

Fourier Transform (FFT). We can see that DAG achieves

the most similar distribution pattern with the flooding

pattern.

Table II and Table III further list the similarity scores

between Flooding measure and one of the three measures

under different similarity functions: Person coefficient, L2

norm and cosine angle. It is clear to see that DAG shows

the best performance compared to GTW and STW, for

example, at KAIST, it improves the accuracy of two

models (GTW and STW) for estimating the centrality of

nodes by 60% and 40% at KAIST, and even by 60% and

100% at Statefair with the L2 norm. The reason behind

this is that it exploits two factors (decayed and weighted)

to differentiate the contact events when modeling the

relationship between two nodes, instead of purely

recording whether there happens a contact event between

them or not.

(a) Flooding versus DAG

(b) Flooding versus GTW

(c) Flooding versus STW

Fig. 4. Fast Fourier Transform (FFT) of centrality distribution at Statefair.

217



TABLE II: SIMILARITY BETWEEN FLOODING AND ONE OF THE THREE

MODELS

(at KAIST) Person L2 Cosine

Flooding vs DAG 0.8664 1.0390 0.9639

Flooding vs GTW 0.5810 1.6633 0.8653

Flooding vs STW 0.8066 1.4797 0.9486

TABLE III: SIMILARITY BETWEEN FLOODING AND ONE OF THE THREE

MODELS

(at Statefair) Person L2 Cosine

Flooding vs DAG 0.8606 0.4543 0.9737

Flooding vs GTW 0.5263 0.7238 0.9324

Flooding vs STW 0.3250 0.9048 0.8984

We now analyze the impact of centrality metric on

worm diffusion speed. To do so, we first calculate the

worm spreading time for each trace including all nodes.

We then repeat the same experiment by removing the

central nodes. We define a central node as a node that

belongs to the top 10% nodes with the highest importance

in the network, the remaining nodes are called non-

central nodes (similar selection/definition has been used

in [24]). Table IV presents the results. The first

phenomenon is that the central nodes play a big role in

worm spreading, removing the central nodes obviously

slows down the diffusion process. The second and

expected phenomenon is that the central nodes selected

from DAG model are responsible for most of the worm

spreading in opportunistic networks. For example,

removing them increases the diffusion time almost by

70% at Statefair, whereas, removing the nodes evaluated

with the other two models only increase the diffusion

time by 27% and 32%, respectively. These results verify

the effectiveness of DAG model.

TABLE IV: WORM DIFFUSION SPEED BY REMOVING THE CENTRAL

NODES EVALUATED WITH THE THREE MODELS

(at) Including

ALL DAG GTW STW

Statefair (s) 130.0 220.0 165.0 172.0

KAIST (s) 311.0 433.0 331.0 383.0

C. Evaluating Similarity

We then explore the impact of the three models (DAG,

GTW and STW) on evaluating similarity between nodes.

Similarity reflects the associations between nodes in the

network. Sociologists have observed the phenomenon

long before, which is called “clustering” in physics, that

if two people have one or more common friends, they can

also be friends with high probability. In general, we use

the min-max function to measure the number of common

neighbors between two nodes.

Let uvSim denote their similarity score, we have

1

min( , )

max( , )

Nui vi

uv

i ui vi

x xSim

x x

(7)

By using the similarity metric, we present another

mobile application where the similarity between nodes

has a heavy influence on the forwarding algorithms in

opportunistic networks [25]-[27]. We employ this

application to observe which of three models best

characterizes the social relationship between nodes, and

at the same time, we also evaluate the impact of similarity

metrics revealed by the three models on the performance

of opportunistic forwarding.

During the forwarding process, if a node u carrying

message m meets up node v, node u forwards the message

to node v and deletes the message from its buffer (i.e., a

single-copy forwarding strategy) if ud vdSim Sim ,where

d denotes the destination of m . The intuition behind this

forwarding strategy is that nodes having high similarity

scores with the destination will deliver the message more

quickly than those with low metrics. We test this intuition

by utilizing the aforementioned two real data-sets. For

each data-set, total 1000 messages are generated and the

emulation results are averaged for consistency.

(a) Statefair

(b) KAIST

Fig. 5. Packet delivery ratio (PDR).

Fig. 5 and Table V show the results on packet delivery

ratio (PDR) and mean delivery delay, two important

performance criteria for opportunistic social networks.

The criterion of packet delivery ratio represents the

delivery reliability in the network in terms of the number

of successfully received messages over that the total sent

messages, and the criterion of mean delivery delay

denotes the delivery efficiency, that is, how much time

we should spend in delivering a message. We can see

that DAG considerably quickens the delivery speed.

Compared to GTW and STW, it improves the delivery

delay by 65% and 52% at Statefair, and by 75% and 28%

at KAIST, respectively, while achieving better results on

packet delivery ratio as well.

218



TABLE V: MEAN DELIVERY DELAY UNDER THE THREE MODELS

(at) DAG GTW STW

Statefair (s) 565.0 934.0 856.0

KAIST (S) 724.0 1266.0 931.0

VI. CONCLUSION AND FUTURE WORK

In this paper, we propose DAG, a decayed aggregation

graph, to model opportunistic social network. Using this

model, we can not only trace the relationship between

nodes over time but also reveal the social features

underlying the OSNs at a fine-grained level. We also

prove that DAG has the same space complexity as the

binary graph with sliding time window. We validate

DAG on two real opportunistic scenarios. The experiment

results show that DAG is effective in characterizing the

relationship between nodes and in improving the

performance of mobile applications. The significant

topics for future work include validating DAG with other

important social features such as the overlapping

community structure, and implementing it into other

crucial applications including recommendation system etc.

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Peiyan Yuan is an associate professor of

Computer Science at the Henan Normal

University since 2014, where he received his

B.S. degree at Computer Science in 2001.

After that, he got his M.S. (2007) and Ph.D.

(2014) degree at Computer Science from

Wuhan University of Technology and Beijing

University of Posts and Telecommunications,

respectively. His research interests include

future networks and distributed systems ranging from content-centric

networks, mobile opportunistic networks to online social networks and

crowd sensing applications etc. He is a member of the CCF and ACM.

219



Yali Wang received her B.S. degree in

computer science from Henan Normal

University, China, and M.S. degree from

Soochow University, China. She is currently a

Ph.D. candidate in computer science and

technology at the State Key Laboratory of

Networking and Switching Technology,

BUPT. Her research interests are in the areas

of Network architecture and Network services.

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Modeling Opportunistic Social Networks with Decayed ... · Modeling Opportunistic Social Networks with Decayed Aggregation Graph . Peiyan Yuan1, 2 and Yali Wang1. 1School of Computer