influence propagation in large graphs - theorems and algorithms

Influence propagation in large graphs -

theorems and algorithmsB. Aditya Prakash

http://www.cs.cmu.edu/~badityap

Christos Faloutsoshttp://www.cs.cmu.edu/~christos

Carnegie Mellon UniversityAUTH’12

Networks are everywhere!

Human Disease Network [Barabasi 2007]

Gene Regulatory Network [Decourty 2008]

Facebook Network [2010]

The Internet [2005]

Prakash and Faloutsos 2012 2AUTH '12

Dynamical Processes over networks are also everywhere!

Prakash and Faloutsos 2012 3AUTH '12

Why do we care?• Information Diffusion• Viral Marketing• Epidemiology and Public Health• Cyber Security• Human mobility • Games and Virtual Worlds • Ecology• Social Collaboration........

Prakash and Faloutsos 2012 4AUTH '12

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

Prakash and Faloutsos 2012 5AUTH '12

Why do we care? (1: Epidemiology)

• 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?

Prakash and Faloutsos 2012 6AUTH '12

Why do we care? (1: Epidemiology)

CURRENT PRACTICE OUR METHOD

~6x fewer!

~6x fewer!

[US-MEDICARE NETWORK 2005]

Prakash and Faloutsos 2012 7AUTH '12Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)

Why do we care? (2: Online Diffusion)

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

~100m active users

> 50m users

Prakash and Faloutsos 2012 8AUTH '12

Why do we care? (2: Online Diffusion)

• Dynamical Processes over networks

Celebrity

Buy Versace™!

Followers

Social Media MarketingPrakash and Faloutsos 2012 9AUTH '12

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

Prakash and Faloutsos 2012 10AUTH '12

Research Theme

DATALarge real-world

networks & processes

ANALYSISUnderstanding

POLICY/ ACTIONManaging

Prakash and Faloutsos 2012 11AUTH '12

In this talk

ANALYSISUnderstanding

Given propagation models:

Q1: Will an epidemic happen?

Prakash and Faloutsos 2012 12AUTH '12

In this talk

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

POLICY/ ACTIONManaging

Prakash and Faloutsos 2012 13AUTH '12

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 14AUTH '12

A fundamental questionStrong Virus

Strong Virus

Epidemic?Epidemic?

Prakash and Faloutsos 2012 15AUTH '12

example (static graph)

Weak VirusWeak Virus

Epidemic?Epidemic?

Prakash and Faloutsos 2012 16AUTH '12

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?

Prakash and Faloutsos 2012 17AUTH '12

Threshold (static version)

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

• Find: –A condition for virus extinction/invasion

Prakash and Faloutsos 2012 18AUTH '12

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…..

Prakash and Faloutsos 2012 19AUTH '12

Outline

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

• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 20AUTH '12

“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 = 3Prakash and Faloutsos 2012 21AUTH '12

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’

Prakash and Faloutsos 2012 22AUTH '12

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

All are about either:

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

• Specific virus propagation models

• Static graphs

Prakash and Faloutsos 2012 23AUTH '12

Outline

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

• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 24AUTH '12

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 …..

Prakash and Faloutsos 2012 25AUTH '12

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 •on the λ, first eigenvalue of A, and •some constant , determined by the virus propagation model

λ λVPMC

No epidemic if λ *

< 1

VPMCVPMC

Prakash and Faloutsos 2012 26AUTH '12In Prakash+ ICDM 2011 (Selected among best papers).

w/ DeepayChakrabarti

Our thresholds for some models

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

Models Effective Strength (s)

Threshold (tipping point)

SIS, SIR, SIRS, SEIR s = λ .

s = 1 SIV, SEIV s = λ .

(H.I.V.) s = λ .

12

221

vv

v

2121 VVISIPrakash and Faloutsos 2012 27AUTH '12

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

u≈ .k

kA

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

kA

Prakash and Faloutsos 2012 28AUTH '12

Largest Eigenvalue (λ)

λ ≈ 2 λ = N λ = N-1

N = 1000λ ≈ 2 λ= 31.67 λ= 999

better connectivity higher λ

Prakash and Faloutsos 2012 29AUTH '12 N nodes

Examples: Simulations – SIR (mumps)

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

PORTLAND graph: synthetic population, 31 million links, 6 million nodes

Effective StrengthTime ticks

Prakash and Faloutsos 2012 30AUTH '12

Examples: Simulations – SIRS (pertusis)

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

PORTLAND graph: synthetic population, 31 million links, 6 million nodesPrakash and Faloutsos 2012 31AUTH '12

Outline

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

• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 32AUTH '12

λ * < 1VPMC

Graph-based

Model-basedGeneral VPM structure

Topology and stability

See paper See paper for full for full proofproof

33AUTH '12 Prakash and Faloutsos 2012

Outline

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

• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 34AUTH '12

Dynamic Graphs: Epidemic?

adjacency matrixadjacency matrix

8

8

Alternating behaviorsDAY (e.g., work)

Prakash and Faloutsos 2012 35AUTH '12

adjacency matrixadjacency matrix

8

8

Dynamic Graphs: Epidemic?Alternating behaviorsNIGHT

(e.g., home)

Prakash and Faloutsos 2012 36AUTH '12

• SIS model– recovery rate δ– infection rate β

• Set of T arbitrary graphs

Model Description

dayday

N

N nightnight

N

N , weekend…..

Infected

Healthy

XN1

N3

N2

Prob. βProb. β Prob. δ

Prakash and Faloutsos 2012 37AUTH '12

• 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 =

Prakash and Faloutsos 2012 38AUTH '12

Synthetic MIT Reality Mining

log(fraction infected)

Time

BELOW

AT

ABOVE ABOVE

AT

BELOW

Infection-profile

Prakash and Faloutsos 2012 39AUTH '12

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

Our threshold

Our threshold

(log scale)

NO EPIDEMIC

EPIDEMIC

EPIDEMIC

NO EPIDEMIC

Synthetic MIT Reality

Prakash and Faloutsos 2012 40AUTH '12

Outline

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

• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 41AUTH '12

Competing Contagions

iPhone v Android Blu-ray v HD-DVD

42AUTH '12 Prakash and Faloutsos 2012Biological common flu/avian flu, pneumococcal inf etc

A simple model

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

Virus 1 Virus 2

Prakash and Faloutsos 2012 43AUTH '12

Question: What happens in the end?

green: virus 1red: virus 2

Footprint @ Steady State Footprint @ Steady State = ?

Number of Infections

Prakash and Faloutsos 2012 44AUTH '12

ASSUME: Virus 1 is stronger than Virus 2

Question: What happens in the end?

green: virus 1red: virus 2

Number of Infections

Strength Strength

??= Strength Strength

2

Footprint @ Steady State Footprint @ Steady State

45AUTH '12 Prakash and Faloutsos 2012

ASSUME: Virus 1 is stronger than Virus 2

Answer: Winner-Takes-All

green: virus 1red: virus 2

Number of Infections

46AUTH '12 Prakash and Faloutsos 2012

ASSUME: Virus 1 is stronger than Virus 2

Our Result: Winner-Takes-All

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

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!

47AUTH '12 Prakash and Faloutsos 2012In Prakash+ WWW 2012

Real Examples

Reddit v Digg Blu-Ray v HD-DVD

[Google Search Trends data]

48AUTH '12 Prakash and Faloutsos 2012

Outline

• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)

Prakash and Faloutsos 2012 49AUTH '12

?

?

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

k = 2

??

Full Static Immunization

Prakash and Faloutsos 2012 50AUTH '12

Outline

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

Prakash and Faloutsos 2012 51AUTH '12

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’?

Prakash and Faloutsos 2012 52AUTH '12

Proposed vulnerability measure λ

Increasing λ Increasing vulnerability

λ is the epidemic threshold

“Safe” “Vulnerable” “Deadly”

Prakash and Faloutsos 2012 53AUTH '12

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

Eigen-Drop(S) Δ λ = λ - λs

Δ

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

Prakash and Faloutsos 2012 54AUTH '12

(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!

Prakash and Faloutsos 2012 55AUTH '12

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

Prakash and Faloutsos 2012 56AUTH '12In Tong, Prakash+ ICDM 2010

Experiment: Immunization quality

Log(fraction of infected nodes)

NetShield

Degree

PageRank

Eigs (=HITS)Acquaintance

Betweeness (shortest path)

Lower is

better TimePrakash and Faloutsos 2012 57AUTH '12

Outline

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

Prakash and Faloutsos 2012 58AUTH '12

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

Under review

Prakash and Faloutsos 2012 59AUTH '12

Fractional Asymmetric Immunization

Hospital Another Hospital

Drug-resistant Bacteria (like XDR-TB)

Prakash and Faloutsos 2012 60AUTH '12

Fractional Asymmetric Immunization

Hospital Another Hospital

Drug-resistant Bacteria (like XDR-TB)

Prakash and Faloutsos 2012 61AUTH '12

Fractional Asymmetric Immunization

Hospital Another Hospital

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

hospitals saved?

Prakash and Faloutsos 2012 62AUTH '12

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!

~6x fewer!

Prakash and Faloutsos 2012 63AUTH '12

Running Time

≈

Simulations SMART-ALLOC

> 1 week

14 secs

> 30,000x speed-up!

Wall-Clock Time

Lower is better

Prakash and Faloutsos 2012 64AUTH '12

Experiments

K = 200 K = 2000

PENN-NETWORK SECOND-LIFE

~5 x ~2.5 x

Lower is better

Prakash and Faloutsos 2012 65AUTH '12

Acknowledgements

Funding

Prakash and Faloutsos 2012 66AUTH '12

References1. 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.)

2. 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

3. 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

4. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon

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

6. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission

7. Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under Submission

67

http://www.cs.cmu.edu/~badityap/AUTH '12 Prakash and Faloutsos 2012

Analysis Policy/Action Data

Propagation on Large Networks

B. Aditya Prakash Christos Faloutsos

68AUTH '12 Prakash and Faloutsos 2012