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Large Graph Mining: Power Tools and a Practitioner’s guide

Large Graph Mining: Power Tools and a Practitioner’s guide. Task 6: Virus/Influence Propagation Faloutsos, Miller,Tsourakakis CMU. Outline. Introduction – Motivation Task 1: Node importance Task 2: Community detection Task 3: Recommendations Task 4: Connection sub-graphs

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Large Graph Mining: Power Tools and a Practitioner’s guide

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  1. Large Graph Mining:Power Tools and a Practitioner’s guide Task 6: Virus/Influence Propagation Faloutsos, Miller,Tsourakakis CMU Faloutsos, Miller, Tsourakakis

  2. Outline Introduction – Motivation Task 1: Node importance Task 2: Community detection Task 3: Recommendations Task 4: Connection sub-graphs Task 5: Mining graphs over time Task 6: Virus/influence propagation Task 7: Spectral graph theory Task 8: Tera/peta graph mining: hadoop Observations – patterns of real graphs Conclusions Faloutsos, Miller, Tsourakakis

  3. Detailed outline • Epidemic threshold • Problem definition • Analysis • Experiments • Fraud detection in e-bay Faloutsos, Miller, Tsourakakis

  4. Virus propagation • How do viruses/rumors propagate? • Blog influence? • Will a flu-like virus linger, or will it become extinct soon? Faloutsos, Miller, Tsourakakis

  5. Prob. d Prob. b Prob. β The model: SIS • ‘Flu’ like: Susceptible-Infected-Susceptible • Virus ‘strength’ s= b/d Healthy N2 N1 N Infected N3 Faloutsos, Miller, Tsourakakis

  6. Epidemic threshold t of a graph: the value of t, such that if strength s = b / d < t an epidemic can not happen Thus, • given a graph • compute its epidemic threshold Faloutsos, Miller, Tsourakakis

  7. Epidemic threshold t What should t depend on? • avg. degree? and/or highest degree? • and/or variance of degree? • and/or third moment of degree? • and/or diameter? Faloutsos, Miller, Tsourakakis

  8. Epidemic threshold • [Theorem 1] We have no epidemic, if β/δ <τ= 1/ λ1,A Faloutsos, Miller, Tsourakakis

  9. Epidemic threshold • [Theorem 1] We have no epidemic (*), if epidemic threshold recovery prob. β/δ <τ= 1/ λ1,A largest eigenvalue of adj. matrix A attack prob. Proof: [Wang+03] (*) under mild, conditional-independence assumptions Faloutsos, Miller, Tsourakakis

  10. details Beginning of proof Healthy @ t+1: - ( healthy or healed ) - and not attacked @ t Let: p(i , t) = Prob node i is sick @ t+1 1 - p(i, t+1 ) = (1 – p(i, t) + p(i, t) * d ) * Pj (1 – b aji * p(j , t) ) Below threshold, if the above non-linear dynamical system above is ‘stable’ (eigenvalue of Hessian < 1 ) Faloutsos, Miller, Tsourakakis

  11. Epidemic threshold for various networks Formula includes older results as special cases: • Homogeneous networks [Kephart+White] • λ1,A = <k>; τ = 1/<k> (<k> : avg degree) • Star networks (d = degree of center) • λ1,A = sqrt(d); τ = 1/ sqrt(d) • Infinite power-law networks • λ1,A = ∞; τ = 0 ; [Barabasi] Faloutsos, Miller, Tsourakakis

  12. Epidemic threshold • [Theorem 2] Below the epidemic threshold, the epidemic dies out exponentially Faloutsos, Miller, Tsourakakis

  13. Detailed outline • Epidemic threshold • Problem definition • Analysis • Experiments • Fraud detection in e-bay Faloutsos, Miller, Tsourakakis

  14. PL-3 Current prediction vs. previous Number of infected nodes PL-3 The formula’s predictions are more accurate Our Our β/δ β/δ Oregon Star Faloutsos, Miller, Tsourakakis

  15. Experiments (Oregon) b/d > τ(above threshold) b/d = τ(at the threshold) b/d < τ(below threshold) Faloutsos, Miller, Tsourakakis

  16. SIS simulation - # infected nodes vs time above Log - Lin at #inf. (log scale) below Time (linear scale) Faloutsos, Miller, Tsourakakis

  17. SIS simulation - # infected nodes vs time above Log - Lin at #inf. (log scale) below Exponential decay Time (linear scale) Faloutsos, Miller, Tsourakakis

  18. SIS simulation - # infected nodes vs time above Log - Log at #inf. (log scale) below Time (log scale) Faloutsos, Miller, Tsourakakis

  19. SIS simulation - # infected nodes vs time above Log - Log at #inf. (log scale) below Power-law Decay (!) Time (log scale) Faloutsos, Miller, Tsourakakis

  20. extra Detailed outline • Epidemic threshold • Fraud detection in e-bay Faloutsos, Miller, Tsourakakis

  21. extra E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU NetProbe: A Fast and Scalable System for Fraud Detection in Online Auction Networks, S. Pandit, D. H. Chau, S. Wang, and C. Faloutsos (WWW'07), pp. 201-210 Faloutsos, Miller, Tsourakakis

  22. extra E-bay Fraud detection • lines: positive feedbacks • would you buy from him/her? Faloutsos, Miller, Tsourakakis

  23. extra E-bay Fraud detection • lines: positive feedbacks • would you buy from him/her? • or him/her? Faloutsos, Miller, Tsourakakis

  24. extra E-bay Fraud detection - NetProbe Belief Propagation gives: Faloutsos, Miller, Tsourakakis

  25. Conclusions • λ1,A : Eigenvalue of adjacency matrix determines the survival of a flu-like virus • It gives a measure of how well connected is the graph (~ # paths – see Task 7, later) • May guide immunization policies • [Belief Propagation: a powerful algo] Faloutsos, Miller, Tsourakakis

  26. References • D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos, Epidemic Thresholds in Real Networks, in ACM TISSEC, 10(4), 2008 • Ganesh, A., Massoulie, L., and Towsley, D., 2005. The effect of network topology on the spread of epidemics. In INFOCOM. Faloutsos, Miller, Tsourakakis

  27. References (cont’d) • Hethcote, H. W. 2000. The mathematics of infectious diseases. SIAM Review 42, 599–653. • Hethcote, H. W. AND Yorke, J. A. 1984. Gonorrhea Transmission Dynamics and Control. Vol. 56. Springer. Lecture Notes in Biomathematics. Faloutsos, Miller, Tsourakakis

  28. References (cont’d) • Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos, Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint, in SRDS 2003 (pages 25-34), Florence, Italy Faloutsos, Miller, Tsourakakis

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