Virus Propagation on Time-Varying Networks:

Theory and Immunization Algorithms

ECML-PKDD 2010, Barcelona, Spain

B. Aditya Prakash*, Hanghang Tong* ^, Nicholas Valler+, Michalis Faloutsos+, Christos

Faloutsos**Carnegie Mellon University, Pittsburgh USA

+University of California – Riverside USA^ IBM Research, Hawthrone USA

Two fundamental questions

Epidemic!

Strong Virus

Q1: Threshold?

example (static graph)

Weak Virus

Small infectio

n

Q1: Threshold?

Questions…

Q2: Immunizatio

n

Which nodes to

immunize?

?

?

Standard, static graph Simple stochastic framework

◦Virus is ‘Flu-like’ (‘SIS’) Underlying contact-network – ‘who-can-

infect-whom’◦Nodes (people/computers) ◦Edges (links between nodes)

OUR CASE:◦Changes in time – alternating behaviors!◦think day vs night

Our Framework

‘S’ Susceptible (= healthy); ‘I’ Infected No immunity (cured nodes -> ‘S’)

Reminder: ‘Flu-like’ (SIS)

Susceptible Infected

Infected by neighbor

Cured internally

Virus birth rate β Host cure rate δ

SIS model (continued)

Infected

Healthy

XN1

N3

N2Prob. β

Prob. β

Prob. δ

Alternating BehaviorsDAY (e.g., work)

adjacency

matrix

8

8

Alternating BehaviorsNIGHT (e.g., home)

adjacency

matrix

8

8

√Our Framework √SIS epidemic model√Time varying graphs

Problem Descriptions Epidemic Threshold Immunization Conclusion

Outline

SIS model◦ cure rate δ◦ infection rate β

Set of T arbitrary graphs

Formally, given

day

N

Nnigh

t

N

N ….weekend…..

Infected

Healthy

XN1

N3

N2

Prob. βProb. β

Prob. δ

Find…

Q1: Epidemic Threshold:Fast die-out?

Q2: Immunizationbest k? ?

?

above

below

I

t

NO epidemic if

eig (S) = < 1

Q1: Threshold - Main result

Single number! Largest eigenvalue of the

“system matrix ”

NO epidemic if eig (S) = < 1

S =

cure rate

infection rate

……..

adjacency matrix

N

N

day night

Details

Synthetic◦ 100 nodes◦ - Clique - Chain

MIT Reality Mining◦ 104 mobile devices◦ September 2004 – June 2005◦ 12-hr adjacency matrices (day) (night)

Q1: Simulation experiments

‘Take-off’ plots

Synthetic MIT Reality Mining

Footprint (# infected @ steady state)

Our threshol

d Our threshol

d

(log scale)

NO EPIDEMIC

EPIDEMIC EPIDEMIC

NO EPIDEMIC

Time-plots

Synthetic MIT Reality Mining

log(# infected)

Time

BELOW threshold

AT threshold

ABOVE threshold

ABOVE threshold

AT threshold

BELOW threshold

√Motivation√Our Framework √SIS epidemic model√Time varying graphs

√Problem Descriptions√Epidemic Threshold Immunization Conclusion

Outline

Our solution◦reduce (== )

◦goal: max ‘eigendrop’ Δ

Comparison - But : No competing policy We propose and evaluate many policies

Q2: Immunization

Δ = _before - _after

?

?

Lower is better

OptimalGreedy-S

Greedy-DavgA

Time-varying Graphs , SIS (flu-like) propagation model

√ Q1: Epidemic Threshold - < 1 ◦Only first eigen-value of system matrix!

√ Q2: Immunization Policies – max. Δ ◦Optimal◦Greedy-S◦Greedy-DavgA◦etc.

Conclusion….

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

Our threshold

Any questions?