Dynamical Processes on Large Networks B. Aditya Prakash badityap Carnegie Mellon University UMD-College Park, April 26, 2012

Dynamical Processes on Large Networks

B. Aditya Prakashhttp://www.cs.cmu.edu/~badityap

Carnegie Mellon University

UMD-College Park, April 26, 2012

Networks are everywhere!

Human Disease Network [Barabasi 2007]

Gene Regulatory Network [Decourty 2008]

Facebook Network [2010]

The Internet [2005]

3

Dynamical Processes over networks are also everywhere!

4

Why do we care?

• Online Information Diffusion• Viral Marketing• Epidemiology and Public Health• Cyber Security• Human mobility • Games and Virtual Worlds • Ecology........

5

Why do we care? (1: Epidemiology)

• Dynamical Processes over networks[AJPH 2007]

CDC data: Visualization of the first 35 tuberculosis (TB) patients and their 1039 contacts

Diseases over contact networks

6


• Dynamical Processes over networks

• Each circle is a hospital• ~3000 hospitals• More than 30,000 patients transferred

[US-MEDICARE NETWORK 2005]

Problem: Given k units of disinfectant, whom to immunize?

7


CURRENT PRACTICE OUR METHOD

~6x fewer!

[US-MEDICARE NETWORK 2005]

Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)

8

Why do we care? (2: Online Diffusion)

> 800m users, ~$1B revenue [WSJ 2010]

~100m active users

> 50m users

9

Why do we care? (2: Online Diffusion)


Celebrity

Buy Versace™!

Followers

Social Media Marketing

10

Why do we care? (3: To change the world?)


Social networks and Collaborative Action

11

High Impact – Multiple Settings

Q. How to squash rumors faster?

Q. How do opinions spread?

Q. How to market better?

epidemic out-breaks

products/viruses

transmit s/w patches

12

Research Theme

DATALarge real-world

networks & processes

ANALYSISUnderstanding

POLICY/ ACTIONManaging

Research Theme – Public Health

DATAModeling # patient

transfers

ANALYSISWill an epidemic

happen?

POLICY/ ACTION

How to control out-breaks?

Research Theme – Social Media

DATAModeling Tweets

spreading

POLICY/ ACTION

How to market better?

ANALYSIS# cascades in

future?

15

In this talk

ANALYSISUnderstanding

Given propagation models:

Q1: Will an epidemic happen?

16

In this talk

Q2: How to immunize and control out-breaks better?

POLICY/ ACTIONManaging

17

In this talk

DATALarge real-world

networks & processes

Q3: How do #hashtags spread?

18

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)• Learning Models: Twitter (Empirical Studies)• Other

19

A fundamental questionStrong Virus

Epidemic?

20

example (static graph)Weak Virus

Epidemic?

21

Problem Statement

Find, a condition under which– virus will die out exponentially quickly– regardless of initial infection condition

above (epidemic)

below (extinction)

# Infected

time

Separate the regimes?

22

Threshold (static version)

Problem Statement• Given: –Graph G, and –Virus specs (attack prob. etc.)

• Find: –A condition for virus extinction/invasion

23

Threshold: Why important?

• Accelerating simulations• Forecasting (‘What-if’ scenarios)• Design of contagion and/or topology• A great handle to manipulate the spreading– Immunization– Maximize collaboration…..

24

Outline

• Motivation• Epidemics: what happens? (Theory)– Background– Result and Intuition (Static Graphs)– Proof Ideas (Static Graphs)– Bonus 1: Dynamic Graphs– Bonus 2: Competing Viruses

• Action: Who to immunize? (Algorithms)• Learning Models: Twitter (Empirical Studies)• Other

25

“SIR” model: life immunity (mumps)

• Each node in the graph is in one of three states– Susceptible (i.e. healthy)– Infected– Removed (i.e. can’t get infected again)

Prob. β Prob. δ

t = 1 t = 2 t = 3

Background

26

Terminology: continued

• Other virus propagation models (“VPM”)– SIS : susceptible-infected-susceptible, flu-like– SIRS : temporary immunity, like pertussis– SEIR : mumps-like, with virus incubation (E = Exposed)….………….

• Underlying contact-network – ‘who-can-infect-whom’

Background

27

Related Work R. M. Anderson and R. M. May. Infectious Diseases of Humans. Oxford University Press,

1991. A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex Networks.

Cambridge University Press, 2010. F. M. Bass. A new product growth for model consumer durables. Management Science,

15(5):215–227, 1969. D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in

real networks. ACM TISSEC, 10(4), 2008. D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly

Connected World. Cambridge University Press, 2010. A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of

epidemics. IEEE INFOCOM, 2005. Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free

networks and the effective immunization. arXiv:cond-at/0305549 v2, Aug. 6 2003. H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42, 2000. H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer

Lecture Notes in Biomathematics, 46, 1984. J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer

viruses. IEEE Computer Society Symposium on Research in Security and Privacy, 1991. J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE

Computer Society Symposium on Research in Security and Privacy, 1993. R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks.

Physical Review Letters 86, 14, 2001.

……… ……… ………

All are about either:

• Structured topologies (cliques, block-diagonals, hierarchies, random)

• Specific virus propagation models

• Static graphs

Background

28

Outline



29

How should the answer look like?

• Answer should depend on:– Graph– Virus Propagation Model (VPM)

• But how??– Graph – average degree? max. degree? diameter?– VPM – which parameters? – How to combine – linear? quadratic? exponential?

?diameterdavg ?/)( max22 ddd avgavg …..

30

Static Graphs: Our Main Result

• Informally,

•

For, any arbitrary topology (adjacency matrix A) any virus propagation model (VPM) in standard literature

the epidemic threshold depends only 1. on the λ, first eigenvalue of A, and 2. some constant , determined by

the virus propagation model

λVPMC

No epidemic if λ *

< 1

VPMCVPMC

In Prakash+ ICDM 2011 (Selected among best papers).

31

Our thresholds for some models

• s = effective strength• s < 1 : below threshold

Models Effective Strength (s) Threshold (tipping point)

SIS, SIR, SIRS, SEIRs = λ .

s = 1

SIV, SEIV s = λ .

(H.I.V.) s = λ .

12

221

vv

v

2121 VVISI

32

Our result: Intuition for λ

“Official” definition:• Let A be the adjacency

matrix. Then λ is the root with the largest magnitude of the characteristic polynomial of A [det(A – xI)].

• Doesn’t give much intuition!

“Un-official” Intuition • λ ~ # paths in the

graph

uu≈ .

kkA

(i, j) = # of paths i j of length k

kA

33

better connectivity higher λ

Largest Eigenvalue (λ)

34N nodes

Largest Eigenvalue (λ)

λ ≈ 2 λ = N λ = N-1

N = 1000λ ≈ 2 λ= 31.67 λ= 999

better connectivity higher λ

35

Examples: Simulations – SIR (mumps)

(a) Infection profile (b) “Take-off” plot

PORTLAND graph31 million links, 6 million nodes

Frac

tion

of In

fecti

ons

Foot

prin

tEffective StrengthTime ticks

36

Examples: Simulations – SIRS (pertusis)

Frac

tion

of In

fecti

ons

Foot

prin

tEffective StrengthTime ticks

(a) Infection profile (b) “Take-off” plot

PORTLAND graph31 million links, 6 million nodes

37

Outline



38

λ * < 1VPMC

Graph-based

Model-based

Proof Sketch

General VPM structure

Topology and stability

39

Models and more modelsModel Used for

SIR Mumps

SIS Flu

SIRS Pertussis

SEIR Chicken-pox

……..

SICR Tuberculosis

MSIR Measles

SIV Sensor Stability

H.I.V.……….

2121 VVISI

40

Ingredient 1: Our generalized model

Endogenous Transitions

Susceptible Infected

Vigilant

Exogenous Transitions




Vigilant

41

Special case


Vigilant

42

Special case: H.I.V.

2121 VVISI

Multiple Infectious, Vigilant states

“Terminal”

“Non-terminal”

43

Ingredient 2: NLDS+Stability

• View as a NLDS– discrete time – non-linear dynamical system (NLDS)

Probability vector Specifies the state of the system at time t

Details

size mN x 1

.

.

.

.

.

size N (number of nodes in the graph)

.

.

.

S

I

V

44

Ingredient 2: NLDS + Stability


Non-linear functionExplicitly gives the evolution of system

Details

size mN x 1

.

.

.

.

.

.

.

.

45

Ingredient 2: NLDS + Stability


• Threshold Stability of NLDS

46

= probability that node i is not attacked by any of its infectious neighbors

Special case: SIR

size 3N x 1 I

R

S

NLDS

I

R

S

Details

47

Fixed Point

11.

00.

00.

State when no node is infected

Q: Is it stable?

Details

48

Stability for SIR

Stableunder threshold

Unstableabove threshold

49

λ * < 1VPMC

Graph-based

Model-basedGeneral VPM structure

Topology and stability

See paper for full proof

50

Outline



51

Dynamic Graphs: Epidemic?

adjacency matrix

8

8

Alternating behaviorsDAY (e.g., work)

52

adjacency matrix

8

8

Dynamic Graphs: Epidemic?Alternating behaviorsNIGHT

(e.g., home)

53

• SIS model– recovery rate δ– infection rate β

• Set of T arbitrary graphs

Model Description

day

N

N night

N

N , weekend…..

Infected

Healthy

XN1

N3

N2

Prob. βProb. β Prob. δ

54

• Informally, NO epidemic if

eig (S) = < 1

Our result: Dynamic Graphs Threshold

Single number! Largest eigenvalue of The system matrix S

In Prakash+, ECML-PKDD 2010

S =

Details

55

Synthetic MIT Reality Mining

log(fraction infected)

Time

BELOW

AT

ABOVE ABOVE

AT

BELOW

Infection-profile

56

“Take-off” plotsFootprint (# infected @ “steady state”)

Our threshold

Our threshold

(log scale)

NO EPIDEMIC

EPIDEMIC

EPIDEMIC

NO EPIDEMIC

Synthetic MIT Reality

57

Outline



58

Competing Contagions

iPhone v Android Blu-ray v HD-DVD

Biological common flu/avian flu, pneumococcal inf etc

59

A simple model

• Modified flu-like • Mutual Immunity (“pick one of the two”)• Susceptible-Infected1-Infected2-Susceptible

Virus 1 Virus 2

Details

60

Question: What happens in the end?

green: virus 1red: virus 2

Footprint @ Steady State Footprint @ Steady State = ?

Number of Infections

ASSUME: Virus 1 is stronger than Virus 2

61

Question: What happens in the end?




Strength Strength

??= Strength Strength

2

Footprint @ Steady State Footprint @ Steady State

62

Answer: Winner-Takes-All




63

Our Result: Winner-Takes-All

In Prakash+ WWW 2012

Given our model, and any graph, the weaker virus always dies-out completely

1. The stronger survives only if it is above threshold 2. Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus 2)3. Strength(Virus) = λ β / δ same as before!

Details

64

Real Examples

Reddit v Digg Blu-Ray v HD-DVD

[Google Search Trends data]

65

Outline


66

?

?

Given: a graph A, virus prop. model and budget k; Find: k ‘best’ nodes for immunization (removal).

k = 2

??

Full Static Immunization

67

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)– Full Immunization (Static Graphs)– Fractional Immunization

• Learning Models: Twitter (Empirical Studies)• Other

68

Challenges

• Given a graph A, budget k, Q1 (Metric) How to measure the ‘shield-

value’ for a set of nodes (S)?

Q2 (Algorithm) How to find a set of k nodes with highest ‘shield-value’?

69

Proposed vulnerability measure λ

Increasing λ Increasing vulnerability

λ is the epidemic threshold

“Safe” “Vulnerable” “Deadly”

70

1

9

10

3

4

5

7

8

6

2

9

1

11

10

3

4

56

7

8

2

9

Original Graph Without {2, 6}

Eigen-Drop(S) Δ λ = λ - λs

Δ

A1: “Eigen-Drop”: an ideal shield value

71

(Q2) - Direct Algorithm too expensive!

• Immunize k nodes which maximize Δ λ

S = argmax Δ λ• Combinatorial!• Complexity:– Example: • 1,000 nodes, with 10,000 edges • It takes 0.01 seconds to compute λ• It takes 2,615 years to find 5-best nodes!

72

A2: Our Solution

• Part 1: Shield Value–Carefully approximate Eigen-drop (Δ λ)–Matrix perturbation theory

• Part 2: Algorithm–Greedily pick best node at each step–Near-optimal due to submodularity

• NetShield (linear complexity)–O(nk2+m) n = # nodes; m = # edges

In Tong, Prakash+ ICDM 2010

73

Experiment: Immunization qualityLog(fraction of infected nodes)

NetShield

Degree

PageRank

Eigs (=HITS)Acquaintance

Betweeness (shortest path)

Lower is

better Time

74

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)– Full Immunization (Static Graphs)– Fractional Immunization

• Learning Models: Twitter (Empirical Studies)• Other

75

Fractional Immunization of NetworksB. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos

Submitted to PNAS

Fractional Asymmetric Immunization

Hospital Another Hospital

76

Drug-resistant Bacteria (like XDR-TB)



Drug-resistant Bacteria (like XDR-TB)

77

= f



78

Problem: Given k units of disinfectant, how to distribute them to maximize

hospitals saved?

79

Our Algorithm “SMART-ALLOC”

CURRENT PRACTICE SMART-ALLOC

[US-MEDICARE NETWORK 2005]• Each circle is a hospital, ~3000 hospitals• More than 30,000 patients transferred

~6x fewer!

Running Time

80

≈

Simulations SMART-ALLOC

> 1 week

14 secs

> 30,000x speed-up!

Wall-Clock Time

Lower is better

81

Experiments

K = 200 K = 2000

PENN-NETWORK SECOND-LIFE

~5 x ~2.5 x

Lower is better

82

Outline


83

Tweets Diffusion: Problem Definition

• Given: – Action log of people tweeting a #hashtag– A network of users

• Find:– How external influence varies with #hashtags?

? ??

?

?? ??

84

Tweet Diffusion: Data

• Yahoo! Twitter firehose• More than 750 million tweets (> 10 Tera-bytes)• Test-bed of > 6000 machines– Hadoop+PIG system ver 0.20.204.0

• Took top 500 hashtags (by volume) in Feb 2011• Network of users:– connecting user X to user Y if X directed at least 3 @-

messages to Y (or RT-ed a tweet)

85

Tweet Diffusion

• Propagation = Influence + External• Developed a model– takes the previous observations into account– with parameters representing external influence

• Learn from previous data– EM-style alternating minimizing algorithm

• Group tags according to learnt params

86

Results: External Influence vs Time

time

“External Effects”

#nowwatching, #nowplaying, #epictweets

#purpleglasses, #brits, #famouslies

#oscar, #25jan

#openfollow, #ihatequotes, #tweetmyjobs

Can also use for Forecasting, Anomaly Detection!

Bursty, external events

“Word-of-mouth” Not trending

Long-running tags

“Word-of-mouth”

87

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)• Learning Models: Twitter (Empirical Studies)• Other – Chipping communities– Time-series analysis

Mobile Call Graphs

88

200K usersMillions of calls

In Prakash+ PAKDD 2010

How do these graphs look like?

• Chain?

• Collection of Stars?

• Cliques+Stars?

89

How do these graphs look like?

• Use Singular(eigen)-vectors?– Intuition: Encode connectivity patterns

0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 ……

≈ . 0 0 0 1 1 1 0 0 90

From

ToAdjacency

MatrixV

Σ V^T

Dimensionality Reduction

EigenPlots

• Plot Eigenvectors side-by-side (EE-plot)

91

V1First Principal Component (Score)

V2 Second Principal Component (Score)

V1 Score

Smith

Johnson

Smith

Johnson

V Matrix

V2 Score

Mock Quiz: EE-plots in our real graph?

• Plot Eigenvectors side-by-side (EE-plots)

92

Spokes!

? ?

??

93

EigenSpokes: they are everywhere

• Mobile call graphs–multiple regions and different months

• And in diverse graphs–Patents citation– Internet–Dictionary

94

EigenSpokes: Reason

• Loosely connected– Near Cliques– Near Bi-partite cores

Core

“Communities”

(near) cliques

(near) bi-cores

95

EigenSpokes: Usefulness?

• Help with community-detection!*– extract nodes with high scores– similar connectivity

Spy Plots

of Top

20 nodes

*http://www.cs.cmu.edu/~badityap/code/spoken.tar

“SpokEn” in action!

96

magnified bipartite community

patents fromsame inventor(s)

cut-and-pastebibliography!

Patent Graph

All patents on photosensitive pigments for color printers

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)• Learning Models: Twitter (Empirical Studies)• Other – Chipping communities– Time-series analysis

97

BGP router updates

• Datapository.net• Abilene Network• 18 million updates over two years!

98

Find patterns and anomalies

BGP-lens at Work

Event 1: – Prefix and Origin-AS pointed to Alabama Supercomputing Net.– Sysadmin :

“the route for 207.157.115.0/24 was appearing and disappearing in [the] IGP routing table ... [which] may have caused BGP to flap.”

– Anomaly went undetected and unresolved for 30 days!

99In Prakash+ SIGKDD 2009

100

Primary and middle Schools in Guangzhou, China.

May 12 – 8 hr spike

Results from real data – Prolonged Spikes

101

Low

High

High energy

Low energy

‘Tornado’doesn’ttouch down

time ->

E2

Frequency Analysis – Wavelet Scalogram

Fake Real

Answering Similarity Queries

102

SELECT * FROM dbWHERE TimeSeriesLIKE “ “

db

In Li and Prakash ICML 2011

• BGP • Health-care• Datacenter Monitoring• Motion-capture……..

Answering Similarity Queries

103

• Euclidean?• DTW? ……

Our Method

• Complex Linear Dynamical Systems (Kalman Filters)

104

Properties Learns joint dynamics Interpretable Features Time Shifts Frequency Proximity Scalable Includes PCA, DFT, AR as special cases

Details

Times Series Analysis

105

BGP data: PLiF + hierarchical clustering

106

Outline


ML & Stats.

Comp. Systems

Theory & Algo.

Biology

Econ.

Social Science

Physics

107

Dynamical Processes

on Networks

Publications1. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni

Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon2. Threshold Conditions for Arbitrary Cascade Models on Arbitrary Networks (B. Aditya Prakash, Deepayan

Chakrabarti, Michalis Faloutsos, Nicholas Valler, Christos Faloutsos) - In IEEE ICDM 2011, Vancouver (Invited to KAIS Journal Best Papers of ICDM.)

3. Times Series Clustering: Complex is Simpler! (Lei Li, B. Aditya Prakash) - In ICML 2011, Bellevue4. Epidemic Spreading on Mobile Ad Hoc Networks: Determining the Tipping Point (Nicholas Valler, B. Aditya

Prakash, Hanghang Tong, Michalis Faloutsos and Christos Faloutsos) – In IEEE NETWORKING 2011, Valencia, Spain

5. Formalizing the BGP stability problem: patterns and a chaotic model (B. Aditya Prakash, Michalis Faloutsos and Christos Faloutsos) – In IEEE INFOCOM NetSciCom Workshop, 2011.

6. On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia

7. Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms (B. Aditya Prakash, Hanghang Tong, Nicholas Valler, Michalis Faloutsos and Christos Faloutsos) – In ECML-PKDD 2010, Barcelona, Spain

8. MetricForensics: A Multi-Level Approach for Mining Volatile Graphs (Keith Henderson, Tina Eliassi-Rad, Christos Faloutsos, Leman Akoglu, Lei Li, Koji Maruhashi, B. Aditya Prakash and Hanghang Tong) - In SIGKDD 2010, Washington D.C.

9. Parsimonious Linear Fingerprinting for Time Series (Lei Li, B. Aditya Prakash and Christos Faloutsos) - In VLDB 2010, Singapore

10. EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs (B. Aditya Prakash, Ashwin Sridharan, Mukund Seshadri, Sridhar Machiraju and Christos Faloutsos) – In PAKDD 2010, Hyderabad, India

11. BGP-lens: Patterns and Anomalies in Internet-Routing Updates (B. Aditya Prakash, Nicholas Valler, David Andersen, Michalis Faloutsos and Christos Faloutsos) – In ACM SIGKDD 2009, Paris, France.

12. Surprising Patterns and Scalable Community Detection in Large Graphs (B. Aditya Prakash, Ashwin Sridharan, Mukund Seshadri, Sridhar Machiraju and Christos Faloutsos) – In IEEE ICDM Large Data Workshop 2009, Miami

13. FRAPP: A Framework for high-Accuracy Privacy-Preserving Mining (Shipra Agarwal, Jayant R. Haritsa and B. Aditya Prakash) – In Intl. Journal on Data Mining and Knowledge Discovery (DKMD), Springer, vol. 18, no. 1, February 2009, Ed: Johannes Gehrke.

14. Complex Group-By Queries For XML (C. Gokhale, N. Gupta, P. Kumar, L. V. S. Lakshmanan, R. Ng and B. Aditya Prakash) – In IEEE ICDE 2007, Istanbul, Turkey.

****

**

**

109

Submitted1. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna,

Hanghang Tong, Christos Faloutsos)

2. How much of Twitter is Influence? (B. Aditya Prakash, Deepayan Chakrabarti, Kunal Punera)

3. Who is to blame? Finding Culprits in Epidemics (B. Aditya Prakash, Jilles Vreeken, Christos Faloutsos)

4. Competing Viruses on Composite Networks: Who wins? (Xuetao Wei, Nicholas Valler, B. Aditya Prakash, Iulian Neamtiu, Michalis Faloutsos and Christos Faloutsos)

5. Gelling, and Melting, Large Graphs through Edge Manipulation (Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad, Michalis Faloutsos, Christos Faloutsos)

6. Worst-case Footprints in the SIS model (B. Aditya Prakash, Varun Gupta and Christos Faloutsos)

Patents7. Determining User Communities in Communication Networks (Ashwin Sridharan, Mukund

Seshadri, James Schneider, B. Aditya Prakash, Christos Faloutsos) Sprint Inc., filed March 2010

8. Analysis of Computer Network Activity by Successively Removing Accepted Types of Access Events (B. Aditya Prakash, Alice Zheng, Jack Stokes, Eric Fitzgerald, Theodore Hardy) Microsoft Research, filed April 2010

**

110

Acknowledgements

Collaborators Christos Faloutsos Roni Rosenfeld, Michalis Faloutsos, Lada Adamic, Theodore Iwashyna (M.D.), Dave Andersen, Tina Eliassi-Rad, Iulian Neamtiu,

Varun Gupta, Jilles Vreeken,

Deepayan Chakrabarti, Hanghang Tong, Kunal Punera, Ashwin Sridharan, Sridhar Machiraju, Mukund Seshadri, Alice Zheng, Lei Li, Polo Chau, Nicholas Valler, Alex Beutel, Xuetao Wei

111

Acknowledgements

Funding

112

Analysis Policy/Action Data

Dynamical Processes on Large Networks

B. Aditya Prakash http://www.cs.cmu.edu/~badityap

Documents

Dynamical Processes on Large Networks B. Aditya Prakash badityap Carnegie Mellon University UMD-College Park, April 26, 2012