1. 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 – March 28

2. Preaching to the choir: Networks are everywhere! Human Disease Network [Barabasi 2007] Gene Regulatory Network [Decourty 2008] Facebook Network [2010] The Internet [2005] Prakash and Faloutsos 20122

3. Focus of this talk: Dynamical Processes over networks are also everywhere! Prakash and Faloutsos 20123

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 20124

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 20125

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 20126

7. Why do we care? (1: Epidemiology) CURRENT PRACTICEOUR METHOD ~6x fewer! [US-MEDICARE NETWORK 2005] Hospital-acquired inf. took 99K+ lives, cost $5B+ (all per year) Prakash and Faloutsos 20127

8. Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m users Prakash and Faloutsos 20128

9. Why do we care? (2: Online Diffusion) Dynamical Processes over networks Celebrity Buy Versace™! Followers Social Media Marketing Prakash and Faloutsos 20129

10. Why do we care? (3: To change the world?) Dynamical Processes over networks Social networks and Collaborative Action Prakash and Faloutsos 201210

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 201211

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

13. Research Theme – Public Health DATA Modeling # patient transfers ANALYSIS Will an epidemic happen? POLICY/ ACTION How to control out-breaks? Prakash and Faloutsos 201213

14. Research Theme – Social Media DATA Modeling Tweets spreading POLICY/ ACTION How to market better? ANALYSIS # cascades in future? Prakash and Faloutsos 201214

15. In this talk ANALYSIS Understanding Given propagation models: Q1: Will an epidemic happen? Prakash and Faloutsos 201215

16. In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION Managing Prakash and Faloutsos 201216

17. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201217

18. A fundamental question Strong Virus Epidemic? Prakash and Faloutsos 201218

19. example (static graph) Weak Virus Epidemic? Prakash and Faloutsos 201219

20. 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 201220

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 201221

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 201222

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 201223

24. “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 201224

25. 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 201225

26. 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 201226

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 201227

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 201228

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 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 In Prakash+ ICDM 2011 (Selected among best papers). Prakash and Faloutsos 201229

30. 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 201230

31. 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 201231

32. better connectivity higher λ Largest Eigenvalue (λ) Prakash and Faloutsos 201232

33. N nodes Largest Eigenvalue (λ) λ ≈ 2λ = Nλ = N-1 N = 1000 λ ≈ 2λ= 31.67λ= 999 better connectivity higher λ Prakash and Faloutsos 201233

34. Examples: Simulations – SIR (mumps) (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Fraction of Infections Footprint Effective Strength Time ticks Prakash and Faloutsos 201234

35. Examples: Simulations – SIRS (pertusis) Fraction of Infections Footprint Effective StrengthTime ticks (a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, 31 million links, 6 million nodes Prakash and Faloutsos 201235

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 201236

37. λ * < 1 Graph-based Model-based General VPM structure Topology and stability See paper for full proof Prakash and Faloutsos 201237

38. 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 201238

39. Dynamic Graphs: Epidemic? adjacency matrix 8 8 Alternating behaviors DAY (e.g., work) Prakash and Faloutsos 201239

40. adjacency matrix 8 8 Dynamic Graphs: Epidemic? Alternating behaviors NIGHT (e.g., home) Prakash and Faloutsos 201240

41. 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 201241

42. 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 201242

43. Synthetic MIT Reality Mining log(fraction infected) Time BELOW AT ABOVE AT BELOW Infection-profile Prakash and Faloutsos 201243

44. “Take-off” plots Footprint (# infected @ “steady state”) Our threshold (log scale) NO EPIDEMIC EPIDEMIC NO EPIDEMIC SyntheticMIT Reality Prakash and Faloutsos 201244

45. 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 201245

46. Competing Contagions iPhone v Android Blu-ray v HD-DVD Biological common flu/avian flu, pneumococcal inf etc 46 AttackRetreat v

47. A simple model Modified flu-like Mutual Immunity (“pick one of the two”) Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2 Prakash and Faloutsos 201247

48. Question: What happens in the end? green: virus 1 red: virus 2 Footprint @ Steady State = ? Number of Infections ASSUME: Virus 1 is stronger than Virus 2 Prakash and Faloutsos 201248

49. Question: What happens in the end? green: virus 1 red: virus 2 Number of Infections ASSUME: Virus 1 is stronger than Virus 2 Strength ?? = Strength 2 Footprint @ Steady State 49

50. Answer: Winner-Takes-All green: virus 1 red: virus 2 ASSUME: Virus 1 is stronger than Virus 2 Number of Infections 50

51. Our Result: Winner-Takes-All In Prakash+ WWW 2012 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! 51

52. Real Examples Reddit v DiggBlu-Ray v HD-DVD [Google Search Trends data] 52

53. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) Prakash and Faloutsos 201253

54. ? ? 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 201254

55. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization Prakash and Faloutsos 201255

56. 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 201256

57. Proposed vulnerability measure λ Increasing λ Increasing vulnerability λ is the epidemic threshold “Safe”“Vulnerable”“Deadly” Prakash and Faloutsos 201257

58. 1 9 10 3 4 5 7 8 6 2 9 1 11 10 3 4 5 6 7 8 2 9 Original Graph Without {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 201258

59. (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 201259

60. 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 In Tong, Prakash+ ICDM 2010 Prakash and Faloutsos 201260

61. Experiment: Immunization quality Log(fraction of infected nodes) NetShield Degree PageRank Eigs (=HITS) Acquaintance Betweeness (shortest path) Lower is better Time Prakash and Faloutsos 201261

62. Outline Motivation Epidemics: what happens? (Theory) Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional Immunization Prakash and Faloutsos 201262

63. Fractional Immunization of Networks B. Aditya Prakash, Lada Adamic, Theodore Iwashyna (M.D.), Hanghang Tong, Christos Faloutsos Submitted to Science Prakash and Faloutsos 201263

64. Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) Prakash and Faloutsos 201264

65. Fractional Asymmetric Immunization Hospital Another Hospital Drug-resistant Bacteria (like XDR-TB) = f Prakash and Faloutsos 201265

66. Fractional Asymmetric Immunization Hospital Another Hospital Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? Prakash and Faloutsos 201266

67. 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 201267

68. Running Time ≈ SimulationsSMART-ALLOC > 1 week 14 secs > 30,000x speed-up! Wall-Clock Time Lower is better Prakash and Faloutsos 201268

69. Acknowledgements Collaborators Christos Faloutsos Roni Rosenfeld, Michalis Faloutsos, Lada Adamic, Theodore Iwashyna (M.D.), Dave Andersen, Tina Eliassi-Rad, Iulian Neamtiu, Varun Gupta, Jilles Vreeken, Deepayan Chakrabarti, Hanghang Tong, Kunal Punera, Ashwin Sridharan, Sridhar Machiraju, Mukund Seshadri, Alice Zheng, Lei Li, Polo Chau, Nicholas Valler, Alex Beutel, Xuetao Wei Prakash and Faloutsos 201269

70. Acknowledgements Funding Prakash and Faloutsos 201270

71. Analysis Policy/Action Data Propagation on Large Networks B. Aditya Prakash Christos Faloutsos Prakash and Faloutsos 201271

72. BACK-UP SLIDES Prakash and Faloutsos 201272

73. λ * < 1 Graph-based Model-based Proof Sketch General VPM structure Topology and stability Prakash and Faloutsos 201273

74. Models and more models ModelUsed for SIRMumps SISFlu SIRSPertussis SEIRChicken-pox …….. SICRTuberculosis MSIRMeasles SIVSensor Stability H.I.V. ………. Prakash and Faloutsos 201274

75. Ingredient 1: Our generalized model Endogenous Transitions SusceptibleInfected Vigilant Exogenous Transitions Endogenous Transitions SusceptibleInfected Vigilant Prakash and Faloutsos 201275

76. Special case SusceptibleInfected Vigilant Prakash and Faloutsos 201276

77. Special case: H.I.V. Multiple Infectious, Vigilant states “Terminal” “Non-terminal” Prakash and Faloutsos 201277

78. Ingredient 2: NLDS+Stability View as a NLDS – discrete time – non-linear dynamical system (NLDS) Probability vector Specifies the state of the system at time t size mN x 1.......... size N (number of nodes in the graph)...... S S I I V V Prakash and Faloutsos 201278

79. Ingredient 2: NLDS + Stability View as a NLDS – discrete time – non-linear dynamical system (NLDS) Non-linear function Explicitly gives the evolution of system size mN x 1................ Prakash and Faloutsos 201279

80. Ingredient 2: NLDS + Stability View as a NLDS – discrete time – non-linear dynamical system (NLDS) Threshold  Stability of NLDS Prakash and Faloutsos 201280

81. = probability that node i is not attacked by any of its infectious neighbors Special case: SIR size 3N x 1 I I R R S S NLDS I I R R S S Prakash and Faloutsos 201281

82. Fixed Point 11.11. 11.11. 00.00. 00.00. 00.00. 00.00. State when no node is infected Q: Is it stable? Prakash and Faloutsos 201282

83. Stability for SIR Stable under threshold Unstable above threshold Prakash and Faloutsos 201283

84. Another special case: SEIV E == exposed latent state ‘Dropping guard’ ‘Pre-emptive vaccination’ SEIV: itself generalizes SIR (mumps) SIS (flu-like) SIRS (pertussis) SEIR (varicella) etc. SEIV: itself generalizes SIR (mumps) SIS (flu-like) SIRS (pertussis) SEIR (varicella) etc. no “sneezes” infectious Prakash and Faloutsos 201284

85. Our Solution: Part 1 Approximate Eigen-drop (Δ λ) Δ λ ≈ SV(S) = – Result using Matrix perturbation theory – u(i) == ‘eigenscore’ ~~ pagerank(i) Au = λ. u u(i) Prakash and Faloutsos 201285

86. P1: node importanceP2: set diversity Original Graph Select by P1Select by P1+P2 Prakash and Faloutsos 201286

87. Our Solution: Part 2: NetShield We prove that: SV(S) is sub-modular ( & monotone non-decreasing ) NetShield: Greedily add best node at each step Corollary: Greedy algorithm works 1. NetShield is near-optimal (w.r.t. max SV(S)) 2. NetShield is O(nk 2 +m) Footnote: near-optimal means SV(S NetShield ) >= (1-1/e) SV(S Opt ) Prakash and Faloutsos 201287

88. > 10 days 0.1 seconds NetShield 100,000 >=1,000,000 10,000 10 1 0.1 0.01 argmax Δ λ argmax SV(S) Time 1,000 100 10,000,000x # of vaccines Lower is better NetShield Speed Prakash and Faloutsos 201288

89. Full Dynamic Immunization Given: Set of T arbitrary graphs Find: k ‘best’ nodes to immunize (remove) day N N night N N, weekend….. In Prakash+ ECML-PKDD 2010 Prakash and Faloutsos 201289

90. Full Dynamic Immunization Our solution – Recall theorem – Simple: reduce (= ) Goal: max eigendrop Δ No competing policy for comparison We propose and evaluate many policies Matrix Product Δ = daynight Prakash and Faloutsos 201290

91. Performance of Policies MIT Reality Mining Lower is better Prakash and Faloutsos 201291

92. Fractional Asymmetric Immunization Fractional Effect Asymmetric Effect Prakash and Faloutsos 201292

93. Fractional Asymmetric Immunization Fractional Effect Asymmetric Effect Prakash and Faloutsos 201293

94. Fractional Asymmetric Immunization Fractional Effect [ f(x) = ] Asymmetric Effect  Edge weakened by half Prakash and Faloutsos 201294

95. Fractional Asymmetric Immunization Fractional Effect [ f(x) = ] Asymmetric Effect   Only incoming edges Prakash and Faloutsos 201295

96. Fractional Asymmetric Immunization Fractional Effect [ f(x) = ] Asymmetric Effect # antidotes = 3 Prakash and Faloutsos 201296

97. Fractional Asymmetric Immunization Fractional Effect [ f(x) = ] Asymmetric Effect # antidotes = 3 Prakash and Faloutsos 201297

98. Fractional Asymmetric Immunization Fractional Effect [ f(x) = ] Asymmetric Effect # antidotes = 3 Prakash and Faloutsos 201298

99. Problem Statement Hospital-transfer networks – Number of patients transferred Given: – The SI model – Directed weighted graph – A total of k antidotes – A weakening function f(x) Find: – the ‘best’ distribution which minimizes the “footprint” at some time t Almost everything is different from before! In Prakash+ 2012 (Submitted to Science) Prakash and Faloutsos 201299

100. Naïve way How to estimate the footprint? – Run simulations? – too slow – takes about 3 weeks for graphs of typical size! Prakash and Faloutsos 2012100

101. Our Solution – Main Idea The SI model has no threshold – any infection will become an epidemic But can bound the expected number of infected nodes at time t Get the distribution which (approx.) minimizes the bound! – Original problem is NP-complete – SMART-ALLOC Greedy and near-optimal if f() is monotone non- increasing convex Prakash and Faloutsos 2012101