1. 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

2. Epidemic! Strong Virus Q1: Threshold?

3. Weak Virus Small infection Q1: Threshold?

4. Q2: Immunization Which nodes to immunize? ? ?

5. 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

6.  ‘S’ Susceptible (= healthy); ‘I’ Infected  No immunity (cured nodes -> ‘S’) SusceptibleInfected Infected by neighbor Cured internally

7.  Virus birth rate β  Host cure rate δ Infected Healthy XN1 N3 N2 Prob. β Prob. δ

8. DAY (e.g., work) DAY (e.g., work) adjacency matrix 8 8

9. NIGHT (e.g., home) NIGHT (e.g., home) adjacency matrix 8 8

10. √Our Framework √SIS epidemic model √Time varying graphs  Problem Descriptions  Epidemic Threshold  Immunization  Conclusion

11.  SIS model ◦ cure rate δ ◦ infection rate β  Set of T arbitrary graphs day N N night N N ….weekend….. Infected Healthy X N1 N3 N2 Prob. β Prob. δ

12. Q1: Epidemic Threshold: Fast die-out? Q1: Epidemic Threshold: Fast die-out? Q2: Immunization best k? Q2: Immunization best k? ? ? above below I t

13.  NO epidemic if eig ( S ) = < 1 Single number! Largest eigenvalue of the “system matrix ” Single number! Largest eigenvalue of the “system matrix ”

14. NO epidemic if eig ( S ) = < 1 S = cure rate infection rate …….. adjacency matrix N N daynight Details

15.  Synthetic ◦ 100 nodes ◦ - Clique - Chain  MIT Reality Mining ◦ 104 mobile devices ◦ September 2004 – June 2005 ◦ 12-hr adjacency matrices (day) (night)

16. SyntheticMIT Reality Mining Footprint (# infected @ steady state) Our threshold (log scale) NO EPIDEMIC EPIDEMIC NO EPIDEMIC

17. SyntheticMIT Reality Mining log(# infected) Time BELOW threshold AT threshold ABOVE threshold AT threshold BELOW threshold

18. √ Motivation √Our Framework √SIS epidemic model √Time varying graphs √Problem Descriptions √Epidemic Threshold  Immunization  Conclusion

19.  Our solution ◦ reduce (== ) ◦ goal: max ‘eigendrop’ Δ  Comparison - But : No competing policy  We propose and evaluate many policies Δ = _before - _after ? ?

20. Lower is better Optimal Greedy-S Greedy-DavgA

21.  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. ….

22.  B. Aditya Prakash http://www.cs.cmu.edu/~badityap Our threshold Any questions?