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Propagation on Large Networks

Explore the impact of dynamical processes on large networks, from epidemiology to online diffusion, in this insightful study by Prakash and Faloutsos. Discover key applications in social collaboration, viral marketing, cyber security, and more. Gain insights into managing data on real-world networks and developing policies for public health and social media. Learn about propagation models, virus epidemics, immunization strategies, and controlling outbreaks effectively. Uncover the significance of threshold conditions for virus extinction and invasion. Dive into the "SIR" model and various virus propagation models to understand network dynamics. Immerse yourself in the world of contagion, topology design, collaboration maximization, and more through this comprehensive analysis of dynamic network processes.

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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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