1 / 28

Virus Propagation Thresholds in Graphs: A Comprehensive Analysis

Learn about epidemic thresholds, virus propagation, and fraud detection in this practitioner's guide to large graph mining. Explore the dynamics of virus spread, computational models, and practical applications. Includes detailed outlines and experimental results.

yardley-pittman
Télécharger la présentation

Virus Propagation Thresholds in Graphs: A Comprehensive Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related