1. Influence propagation in large graphs - theorems and algorithms B. Aditya Prakash http://www.cs.cmu.edu/~badityap Christos Faloutsos http://www.cs.cmu.edu/~christos Carnegie Mellon University GraphEx’12

2. Thank you! Nadya Bliss Lori Tsoulas GraphEx '12Prakash and Faloutsos 20122

3. Networks are everywhere! Human Disease Network [Barabasi 2007] Gene Regulatory Network [Decourty 2008] Facebook Network [2010] The Internet [2005] Prakash and Faloutsos 20123GraphEx '12

4. Dynamical Processes over networks are also everywhere! Prakash and Faloutsos 20124GraphEx '12

5. Why do we care? Information Diffusion Viral Marketing Epidemiology and Public Health Cyber Security Human mobility Games and Virtual Worlds Ecology Social Collaboration........ Prakash and Faloutsos 20125GraphEx '12

6. 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 20126GraphEx '12

7. 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 20127GraphEx '12

8. Why do we care? (1: Epidemiology) CURRENT PRACTICEOUR METHOD ~6x fewer! [US-MEDICARE NETWORK 2005] Prakash and Faloutsos 20128GraphEx '12 Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year)

9. Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users Prakash and Faloutsos 20129GraphEx '12

10. Why do we care? (2: Online Diffusion) Dynamical Processes over networks Celebrity Buy Versace™! Followers Social Media Marketing Prakash and Faloutsos 201210GraphEx '12

11. High Impact – Multiple Settings Q. How to squash rumors faster? Q. How do opinions spread? Q. How to market better? epidemic out-breaks products/viruses transmit s/w patches Prakash and Faloutsos 201211GraphEx '12

12. Research Theme DATA Large real-world networks & processes ANALYSIS Understanding POLICY/ ACTION Managing Prakash and Faloutsos 201212GraphEx '12

13. In this talk ANALYSIS Understanding Given propagation models: Q1: Will an epidemic happen? Prakash and Faloutsos 201213GraphEx '12

14. In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION Managing Prakash and Faloutsos 201214GraphEx '12

15. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201215GraphEx '12

16. A fundamental question Strong Virus Epidemic? Prakash and Faloutsos 201216GraphEx '12

17. example (static graph) Weak Virus Epidemic? Prakash and Faloutsos 201217GraphEx '12

18. Problem Statement Find, a condition under which – virus will die out exponentially quickly – regardless of initial infection condition above (epidemic) below (extinction) # Infected time Separate the regimes? Prakash and Faloutsos 201218GraphEx '12

19. Threshold (static version) Problem Statement Given: – Graph G, and – Virus specs (attack prob. etc.) Find: – A condition for virus extinction/invasion Prakash and Faloutsos 201219GraphEx '12

20. Threshold: Why important? Accelerating simulations Forecasting (‘What-if’ scenarios) Design of contagion and/or topology A great handle to manipulate the spreading – Immunization ….. Prakash and Faloutsos 201220GraphEx '12

21. 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 201221GraphEx '12

22. “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 = 1t = 2t = 3 Prakash and Faloutsos 201222GraphEx '12

23. 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 201223GraphEx '12

24. Related Work  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.  ……… All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propagation models Static graphs All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propagation models Static graphs Prakash and Faloutsos 201224GraphEx '12

25. 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 201225GraphEx '12

26. 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 201226GraphEx '12

27. 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 1.on the λ, first eigenvalue of A, and 2.some constant, determined by the virus propagation model λ λ No epidemic if λ * < 1 Prakash and Faloutsos 201227GraphEx '12 In Prakash+ ICDM 2011 (Selected among best papers). w/ Deepay Chakrabarti

28. Our thresholds for some models s = effective strength s < 1 : below threshold Models Effective Strength (s) Threshold (tipping point) SIS, SIR, SIRS, SEIR s = λ. s = 1 SIV, SEIV s = λ. ( H.I.V. ) s = λ. Prakash and Faloutsos 201228GraphEx '12

29. Our result: Intuition for λ “Official” definition: 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! “Un-official” Intuition λ ~ # paths in the graph u u ≈. (i, j) = # of paths i  j of length k Prakash and Faloutsos 201229GraphEx '12

30. N nodes Largest Eigenvalue (λ) λ ≈ 2λ = Nλ = N-1 N = 1000 λ ≈ 2λ= 31.67λ= 999 better connectivity higher λ Prakash and Faloutsos 201230GraphEx '12

31. Examples: Simulations – SIR (mumps) Fraction of Infections Footprint Effective Strength Time ticks Prakash and Faloutsos 201231GraphEx '12 (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes

32. Examples: Simulations – SIRS (pertusis) Fraction of Infections Footprint Effective StrengthTime ticks Prakash and Faloutsos 201232GraphEx '12 (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes

33. 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 201233GraphEx '12

34. λ * < 1 Graph-based Model-based General VPM structure Topology and stability See paper for full proof 34

35. 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 201235GraphEx '12

36. Dynamic Graphs: Epidemic? adjacency matrix 8 8 Alternating behaviors DAY (e.g., work) Prakash and Faloutsos 201236GraphEx '12

37. adjacency matrix 8 8 Dynamic Graphs: Epidemic? Alternating behaviors NIGHT (e.g., home) Prakash and Faloutsos 201237GraphEx '12

38. SIS model – recovery rate δ – infection rate β Set of T arbitrary graphs Model Description day N N night N N, weekend….. Infected Healthy XN1 N3 N2 Prob. β Prob. δ Prakash and Faloutsos 201238GraphEx '12

39. Informally, NO epidemic if eig (S) = < 1 Our result: Dynamic Graphs Threshold Single number! Largest eigenvalue of The system matrix S In Prakash+, ECML-PKDD 2010 S = Prakash and Faloutsos 201239GraphEx '12

40. Synthetic MIT Reality Mining log(fraction infected) Time BELOW AT ABOVE AT BELOW Infection-profile Prakash and Faloutsos 201240GraphEx '12

41. “Take-off” plots Footprint (# infected @ “steady state”) Our threshold (log scale) NO EPIDEMIC EPIDEMIC NO EPIDEMIC SyntheticMIT Reality Prakash and Faloutsos 201241GraphEx '12

42. 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 201242GraphEx '12

43. Competing Contagions iPhone v AndroidBlu-ray v HD-DVD 43GraphEx '12Prakash and Faloutsos 2012 Biological common flu/avian flu, pneumococcal inf etc

44. A simple model Modified flu-like Mutual Immunity (“pick one of the two”) Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2 Prakash and Faloutsos 201244GraphEx '12

45. Question: What happens in the end? green: virus 1 red: virus 2 Footprint @ Steady State = ? Number of Infections Prakash and Faloutsos 201245GraphEx '12 ASSUME: Virus 1 is stronger than Virus 2

46. Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections Strength ?? = Strength 2 Footprint @ Steady State 46GraphEx '12Prakash and Faloutsos 2012 ASSUME: Virus 1 is stronger than Virus 2

47. Answer: Winner-Takes-All green: virus 1 red: virus 2 Number of Infections 47GraphEx '12Prakash and Faloutsos 2012 ASSUME: Virus 1 is stronger than Virus 2

48. Our Result: Winner-Takes-All Given our model, and any graph, the weaker virus always dies-out completely 1.The stronger survives only if it is above threshold 2.Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus 2) 3.Strength(Virus) = λ β / δ  same as before! 48GraphEx '12Prakash and Faloutsos 2012 In Prakash+ WWW 2012

49. Real Examples Reddit v DiggBlu-Ray v HD-DVD [Google Search Trends data] 49GraphEx '12Prakash and Faloutsos 2012

50. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201250GraphEx '12

51. ? ? Given: a graph A, virus prop. model and budget k; Find: k ‘best’ nodes for immunization (removal). k = 2 ? ? Full Static Immunization Prakash and Faloutsos 201251GraphEx '12

52. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization Prakash and Faloutsos 201252GraphEx '12

53. Challenges Given a graph A, budget k, Q1 (Metric) How to measure the ‘shield- value’ for a set of nodes (S)? Q2 (Algorithm) How to find a set of k nodes with highest ‘shield-value’? Prakash and Faloutsos 201253GraphEx '12

54. Proposed vulnerability measure λ Increasing λ Increasing vulnerability λ is the epidemic threshold “Safe”“Vulnerable”“Deadly” Prakash and Faloutsos 201254GraphEx '12

55. 1 9 10 3 4 5 7 8 6 2 9 1 11 10 3 4 5 6 7 8 2 9 Original GraphWithout {2, 6} Eigen-Drop(S) Δ λ = λ - λ s Eigen-Drop(S) Δ λ = λ - λ s Δ A1: “Eigen-Drop”: an ideal shield value Prakash and Faloutsos 201255GraphEx '12

56. (Q2) - Direct Algorithm too expensive! Immunize k nodes which maximize Δ λ S = argmax Δ λ Combinatorial! Complexity: – Example: 1,000 nodes, with 10,000 edges It takes 0.01 seconds to compute λ It takes 2,615 years to find 5-best nodes ! Prakash and Faloutsos 201256GraphEx '12

57. A2: Our Solution Part 1: Shield Value – Carefully approximate Eigen-drop (Δ λ) – Matrix perturbation theory Part 2: Algorithm – Greedily pick best node at each step – Near-optimal due to submodularity NetShield (linear complexity) – O(nk 2 +m) n = # nodes; m = # edges Prakash and Faloutsos 201257GraphEx '12 In Tong, Prakash+ ICDM 2010

58. Experiment: Immunization quality Log(fraction of infected nodes) NetShield Degree PageRank Eigs (=HITS) Acquaintance Betweeness (shortest path) Lower is better Time Prakash and Faloutsos 201258GraphEx '12

59. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization Prakash and Faloutsos 201259GraphEx '12

60. Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Under review Prakash and Faloutsos 201260GraphEx '12

61. Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) Prakash and Faloutsos 201261GraphEx '12

62. Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) Prakash and Faloutsos 201262GraphEx '12

63. Fractional Asymmetric Immunization Hospital Another Hospital Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? Prakash and Faloutsos 201263GraphEx '12

64. Our Algorithm “SMART- ALLOC” CURRENT PRACTICESMART-ALLOC [US-MEDICARE NETWORK 2005] Each circle is a hospital, ~3000 hospitals More than 30,000 patients transferred ~6x fewer! Prakash and Faloutsos 201264GraphEx '12

65. Running Time ≈ SimulationsSMART-ALLOC > 1 week 14 secs > 30,000x speed-up! Wall-Clock Time Lower is better Prakash and Faloutsos 201265GraphEx '12

66. Experiments K = 200K = 2000 PENN-NETWORK SECOND-LIFE ~5 x ~2.5 x Lower is better Prakash and Faloutsos 201266GraphEx '12

67. Acknowledgements Funding Prakash and Faloutsos 201267GraphEx '12

68. References 1. Threshold Conditions for Arbitrary Cascade Models on Arbitrary Networks (B. Aditya Prakash, Deepayan Chakrabarti, Michalis Faloutsos, Nicholas Valler, Christos Faloutsos) - In IEEE ICDM 2011, Vancouver (Invited to KAIS Journal Best Papers of ICDM.) 2. Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms (B. Aditya Prakash, Hanghang Tong, Nicholas Valler, Michalis Faloutsos and Christos Faloutsos) – In ECML-PKDD 2010, Barcelona, Spain 3. Epidemic Spreading on Mobile Ad Hoc Networks: Determining the Tipping Point (Nicholas Valler, B. Aditya Prakash, Hanghang Tong, Michalis Faloutsos and Christos Faloutsos) – In IEEE NETWORKING 2011, Valencia, Spain 4. Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon 5. On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi- Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia 6. Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission 7. Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under Submission 68 http://www.cs.cmu.edu/~badityap/

69. Analysis Policy/Action Data Propagation on Large Networks B. Aditya Prakash Christos Faloutsos 69