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Propagation on Large Networks

Propagation on Large Networks. B. Aditya Prakash http://www.cs.cmu.edu/~badityap Christos Faloutsos http://www.cs.cmu.edu/~christos Carnegie Mellon University. INARC Meeting – May 2nd. Preaching to the choir: Networks are everywhere!. Facebook Network [2010].

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Propagation on Large Networks

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  1. Propagation on Large Networks B. AdityaPrakash http://www.cs.cmu.edu/~badityap Christos Faloutsos http://www.cs.cmu.edu/~christos Carnegie Mellon University INARC Meeting – May 2nd

  2. Preaching to the choir:Networks are everywhere! Facebook Network [2010] Gene Regulatory Network [Decourty 2008] Human Disease Network [Barabasi 2007] The Internet [2005] Prakash and Faloutsos 2012

  3. Focus of this talk: Dynamical Processes over networks are also everywhere! Prakash and Faloutsos 2012

  4. Why do we care? • Social collaboration • Information Diffusion • Viral Marketing • Epidemiology and Public Health • Cyber Security • Human mobility • Games and Virtual Worlds • Ecology ........ Prakash and Faloutsos 2012

  5. 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

  6. 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

  7. Why do we care? (1: Epidemiology) ~6x fewer! [US-MEDICARE NETWORK 2005] CURRENT PRACTICE OUR METHOD Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year) Prakash and Faloutsos 2012

  8. Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users Prakash and Faloutsos 2012

  9. Why do we care? (2: Online Diffusion) • Dynamical Processes over networks Buy Versace™! Followers Celebrity Social Media Marketing Prakash and Faloutsos 2012

  10. Why do we care? (3: To change the world?) • Dynamical Processes over networks Social networks and Collaborative Action Prakash and Faloutsos 2012

  11. High Impact – Multiple Settings epidemic out-breaks Q. How to squash rumors faster? Q. How do opinions spread? Q. How to market better? products/viruses transmit s/w patches Prakash and Faloutsos 2012

  12. Research Theme ANALYSIS Understanding POLICY/ ACTION Managing DATA Large real-world networks & processes Prakash and Faloutsos 2012

  13. Research Theme – Public Health ANALYSIS Will an epidemic happen? POLICY/ ACTION How to control out-breaks? DATA Modeling # patient transfers Prakash and Faloutsos 2012

  14. Research Theme – Social Media ANALYSIS # cascades in future? POLICY/ ACTION How to market better? DATA Modeling Tweets spreading Prakash and Faloutsos 2012

  15. In this talk Given propagation models: Q1: Will an epidemic happen? ANALYSIS Understanding Prakash and Faloutsos 2012

  16. In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION Managing Prakash and Faloutsos 2012

  17. Outline • Motivation • Epidemics: what happens? (Theory) • Action: Who to immunize? (Algorithms) Prakash and Faloutsos 2012

  18. A fundamental question Strong Virus Epidemic? Prakash and Faloutsos 2012

  19. example (static graph) Weak Virus Epidemic? Prakash and Faloutsos 2012

  20. Problem Statement # Infected above (epidemic) below (extinction) time Separate the regimes? Find, a condition under which • virus will die out exponentially quickly • regardless of initial infection condition Prakash and Faloutsos 2012

  21. 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

  22. 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

  23. 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

  24. Background “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 = 3 Prakash and Faloutsos 2012

  25. Background 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

  26. Background Related Work • All are about either: • Structured topologies (cliques, block-diagonals, hierarchies, random) • Specific virus propagation models • Static graphs • 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. • ……… • ……… • ……… Prakash and Faloutsos 2012

  27. 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

  28. 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? Prakash and Faloutsos 2012

  29. 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,firsteigenvalueof A,and • some constant , determined by the virus propagation model λ • No epidemic if λ * < 1 In Prakash+ ICDM 2011 (Selected among best papers). Prakash and Faloutsos 2012

  30. Our thresholds for some models s = effective strength s < 1 : below threshold Prakash and Faloutsos 2012

  31. Our result: Intuition for λ “Official” definition: “Un-official” Intuition  λ ~ # paths in the graph • 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! u u ≈ . (i, j) = # of paths i j of length k Prakash and Faloutsos 2012

  32. Largest Eigenvalue (λ) better connectivity higher λ Prakash and Faloutsos 2012

  33. Largest Eigenvalue (λ) better connectivity higher λ λ ≈ 2 λ = N λ = N-1 λ ≈ 2 λ= 31.67 λ= 999 N = 1000 N nodes Prakash and Faloutsos 2012

  34. Examples: Simulations – SIR (mumps) Fraction of Infections Footprint (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Effective Strength Time ticks Prakash and Faloutsos 2012

  35. Examples: Simulations – SIRS (pertusis) Fraction of Infections Footprint (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Time ticks Effective Strength Prakash and Faloutsos 2012

  36. 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

  37. Proof Sketch General VPM structure Model-based λ * < 1 Graph-based Topology and stability

  38. Models and more models

  39. Ingredient 1: Our generalized model Endogenous Transitions Endogenous Transitions Susceptible Susceptible Infected Infected Exogenous Transitions Vigilant Vigilant Endogenous Transitions

  40. Special case Susceptible Infected Vigilant

  41. Special case: H.I.V. “Non-terminal” “Terminal” Multiple Infectious, Vigilant states

  42. Details Ingredient 2: NLDS+Stability size N (number of nodes in the graph) S • Probability vector Specifies the state of the system at time t . . . size mNx 1 I V . . . . . • View as a NLDS • discrete time • non-linear dynamical system (NLDS)

  43. Details Ingredient 2: NLDS + Stability Non-linear function Explicitly gives the evolution of system . . . size mNx 1 . . . . . • View as a NLDS • discrete time • non-linear dynamical system (NLDS)

  44. Ingredient 2: NLDS + Stability • View as a NLDS • discrete time • non-linear dynamical system (NLDS) • Threshold  Stability of NLDS

  45. Details Special case: SIR S S size 3Nx1 I I R R = probability that node iis not attacked by any of its infectious neighbors NLDS

  46. Details Fixed Point 1 1 . 0 0 . 0 0 . State when no node is infected Q: Is it stable?

  47. Stability for SIR Stable under threshold Unstable above threshold

  48. General VPM structure Model-based See paper for full proof λ * < 1 Graph-based Topology and stability Prakash and Faloutsos 2012

  49. 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

  50. Dynamic Graphs: Epidemic? Alternating behaviors • DAY • (e.g., work) adjacency matrix 8 8 Prakash and Faloutsos 2012

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