Influence Propagation in Large Graphs - Theorems and Algorithms

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Professor Christos Faloutsos (Carnegie Mellon University, USA), gave a lecture on "Influence Propagation in Large Graphs - Theorems and Algorithms" in the Distinguished Lecturer Series - Leon The Mathematician.

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  • (S*I*V*?)
  • (S*I*V*?)
  • Curly braces…say what the blue yellow are…
  • Apply g on p_T, you get p_t+1
  • Dimensional arguments…
  • Influence Propagation in Large Graphs - Theorems and Algorithms

    1. 1. Influence propagation inlarge graphs - theorems and algorithms B. Aditya Prakash http://www.cs.cmu.edu/~badityap Christos Faloutsos http://www.cs.cmu.edu/~christos Carnegie Mellon University A.U.T.’12
    2. 2. Thank you!• Yannis ManolopoulosAUT 12 Prakash and Faloutsos 2012 2
    3. 3. Networks are everywhere! Facebook Network [2010] Gene Regulatory Network [Decourty 2008] Human Disease Network [Barabasi 2007] The Internet [2005]AUT 12 Prakash and Faloutsos 2012 3
    4. 4. Dynamical Processes over networks are also everywhere!AUT 12 Prakash and Faloutsos 2012 4
    5. 5. Why do we care?• Information Diffusion• Viral Marketing• Epidemiology and Public Health• Cyber Security• Human mobility• Games and Virtual Worlds• Ecology• Social Collaboration........AUT 12 Prakash and Faloutsos 2012 5
    6. 6. Why do we care? (1: Epidemiology)• Dynamical Processes over networks [AJPH 2007] CDC data: Visualization of the first 35Diseases over contact networks tuberculosis (TB) patients and their 1039 contactsAUT 12 Prakash and Faloutsos 2012 6
    7. 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 Problem: Given k units ofNETWORK 2005] disinfectant, whom to immunize? AUT 12 Prakash and Faloutsos 2012 7
    8. 8. Why do we care? (1: Epidemiology) ~6x fewer! [US-MEDICARE NETWORK 2005] CURRENT PRACTICE OUR METHODHospital-acquired inf. took 99K+ lives, cost $5B+ (all per year) AUT 12 Prakash and Faloutsos 2012 8
    9. 9. Why do we care? (2: Online Diffusion) > 800m users, ~$1B revenue [WSJ 2010] ~100m active users > 50m usersAUT 12 Prakash and Faloutsos 2012 9
    10. 10. Why do we care? (2: Online Diffusion) • Dynamical Processes over networks Buy Versace™!Followers Celebrity Social Media Marketing AUT 12 Prakash and Faloutsos 2012 10
    11. 11. High Impact – Multiple Settings epidemic out-breaksQ. How to squash rumors faster? products/virusesQ. How do opinions spread? transmit s/w patchesQ. How to market better?AUT 12 Prakash and Faloutsos 2012 11
    12. 12. Research Theme ANALYSIS Understanding POLICY/ DATA ACTION Large real-world Managingnetworks & processes AUT 12 Prakash and Faloutsos 2012 12
    13. 13. In this talk Given propagation models: Q1: Will an epidemic happen? ANALYSIS UnderstandingAUT 12 Prakash and Faloutsos 2012 13
    14. 14. In this talk Q2: How to immunize and control out-breaks better? POLICY/ ACTION ManagingAUT 12 Prakash and Faloutsos 2012 14
    15. 15. Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)AUT 12 Prakash and Faloutsos 2012 15
    16. 16. A fundamental question Strong Virus Epidemic?AUT 12 Prakash and Faloutsos 2012 16
    17. 17. example (static graph) Weak Virus Epidemic?AUT 12 Prakash and Faloutsos 2012 17
    18. 18. Problem Statement# Infected above (epidemic) Separate the regimes? below (extinction) time Find, a condition under which – virus will die out exponentially quickly – regardless of initial infection condition AUT 12 Prakash and Faloutsos 2012 18
    19. 19. Threshold (static version)Problem Statement• Given: – Graph G, and – Virus specs (attack prob. etc.)• Find: – A condition for virus extinction/invasionAUT 12 Prakash and Faloutsos 2012 19
    20. 20. 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 …..AUT 12 Prakash and Faloutsos 2012 20
    21. 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)AUT 12 Prakash and Faloutsos 2012 21
    22. 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. δ .β Prob t=1 t=2 t=3AUT 12 Prakash and Faloutsos 2012 22
    23. 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’AUT 12 Prakash and Faloutsos 2012 23
    24. 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. All are about either: 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. • Structured 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 topologies (cliques, epidemics. IEEE INFOCOM, 2005. Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free block-diagonals, hierarchies, random) 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. • Specific virus R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, 14, 2001. propagation models ……… ……… • Static graphs ……… AUT 12 Prakash and Faloutsos 2012 24
    25. 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)AUT 12 Prakash and Faloutsos 2012 25
    26. 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?βd avg + δ diameter ? ( β 2 d 2 avg − δd avg ) / d max ? ….. AUT 12 Prakash and Faloutsos 2012 26
    27. 27. Static Graphs: Our Main Result • Informally, w/ Deepay Chakrabarti For,  any arbitrary topology (adjacency matrix A) λ  any virus propagation model (VPM) in standard literature CVPM the epidemic threshold depends only • No 8.on the λ, first eigenvalue of A, and epidemic if 9.some constant CVPM determined by the , CVPM virus propagation modelIn Prakash+ ICDM 2011 (Selected among best AUT 12 Prakash and Faloutsos 2012 27
    28. 28. Our thresholds for some models • s = effective strength • s < 1 : below thresholdModels Effective Strength Threshold (tipping (s) point)SIS, SIR, SIRS, SEIR β  s=λ.   δ   βγ   s=1SIV, SEIV s=λ.  δ (γ +θ )      β1v2 + β2ε SI 1 I 2 V1 V2 (H.I.V.) s = λ .  v2 ( ε + v1 )      AUT 12 Prakash and Faloutsos 2012 28
    29. 29. Our result: Intuition for λ “Official” definition: “Un-official” Intuition • Let A be the adjacency • λ ~ # paths in the graph matrix. Then λ is the root with the largest magnitude of the characteristic polynomial k A of A [det(A – xI)]. ≈λ k . u• Doesn’t give much intuition! k A (i, j) = # of paths i  j of length k AUT 12 Prakash and Faloutsos 2012 29
    30. 30. Largest Eigenvalue (λ) better connectivity higher λ λ≈2 λ= N λ = N-1 λ≈2 λ= 31.67 λ= 999N = 1000 AUT 12 Prakash and Faloutsos 2012 N nodes 30
    31. 31. Examples: Simulations – SIR (mumps) Time ticks Effective Strength(a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, AUT 12 31 million links, 6 million nodes Prakash and Faloutsos 2012 31
    32. 32. Examples: Simulations – SIRS (pertusis) Time ticks Effective Strength(a) Infection profile (b) “Take-off” plot PORTLAND graph: synthetic population, AUT 12 31 million links, 6 million nodes Prakash and Faloutsos 2012 32
    33. 33. Models and more modelsModel Used forSIR MumpsSIS FluSIRS PertussisSEIR Chicken-pox……..SICR TuberculosisMSIR MeaslesSIV Sensor StabilitySI 1 I 2 V1 V2 H.I.V.………. 33
    34. 34. Ingredient 1: Our generalized model Endogenous Transitions Susceptible Infected Exogenous Transitions Endogenous Vigilant Transitions 34
    35. 35. Special caseSusceptible Infected Vigilant 35
    36. 36. Special case: H.I.V.SI 1 I 2 V1 V2 “Non-terminal” “Terminal” Multiple Infectious, Vigilant states 36
    37. 37. Ingredient 2: NLDS+Stability • View as a NLDS – discrete time – non-linear dynamical system (NLDS) S . . Probability vector . Specifies the state IsizemN x 1 of the system at time t size N (number of V . . nodes in the . 37 graph)
    38. 38. Ingredient 2: NLDS + Stability • View as a NLDS – discrete time – non-linear dynamical system (NLDS) . . Non-linear . functionsizemN x 1 Explicitly gives the evolution of system . . . 38
    39. 39. Ingredient 2: NLDS + Stability• View as a NLDS – discrete time – non-linear dynamical system (NLDS)• Threshold  Stability of NLDS 39
    40. 40. Special case: SIR S Ssize I3N x 1 I R R = probability that node i is not attacked by any of its infectious NLDS neighbors 40
    41. 41. Fixed Point11.00 State when no node. is infected00. Q: Is it stable? 41
    42. 42. Stability for SIR Stable Unstableunder threshold above threshold 42
    43. 43. See paper for full proofGeneral VPMstructure Model-based λ * CVPM < 1 Graph-basedTopologyand stability 43
    44. 44. 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)AUT 12 Prakash and Faloutsos 2012 44
    45. 45. See paper for full proof General VPM structure Model-based λ * CVPM < 1 Graph-based Topology and stabilityAUT 12 Prakash and Faloutsos 2012 45
    46. 46. 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)AUT 12 Prakash and Faloutsos 2012 46
    47. 47. Dynamic Graphs: Epidemic?DAY Alternating behaviors(e.g., work)adjacency 8matrix 8AUT 12 Prakash and Faloutsos 2012 47
    48. 48. Dynamic Graphs: Epidemic?NIGHT Alternating behaviors(e.g., home)adjacency 8matrix 8AUT 12 Prakash and Faloutsos 2012 48
    49. 49. Model Description Healthy N2• SIS model Prob. β – recovery rate δ N1 X Prob Prob. δ .β – infection rate β Infected N3• Set of T arbitrary graphs day N N night , weekend….. N NAUT 12 Prakash and Faloutsos 2012 49
    50. 50. Our result: Dynamic Graphs Threshold • Informally, NO epidemic if eig (S) = <1 Single number! S =Largest eigenvalue of The system matrix SIn Prakash+, ECML-PKDD 2010 AUT 12 Prakash and Faloutsos 2012 50
    51. 51. Infection-profilelog(fraction infected) Synthetic MIT Reality Mining ABOVE ABOVE AT AT BELOW BELOW Time AUT 12 Prakash and Faloutsos 2012 51
    52. 52. Footprint (#infected @“steady state”) “Take-off” plots Synthetic MIT Reality EPIDEMIC Our EPIDEMIC Our threshold threshold NO EPIDEMIC NO EPIDEMIC (log scale) AUT 12 Prakash and Faloutsos 2012 52
    53. 53. 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)AUT 12 Prakash and Faloutsos 2012 53
    54. 54. Competing Contagions iPhone v Android Blu-ray v HD-DVDBiological common flu/avian flu, pneumococcal inf AUT 12 Prakash and Faloutsos 2012 54
    55. 55. A simple model• Modified flu-like• Mutual Immunity (“pick one of the two”)• Susceptible-Infected1-Infected2-Susceptible Virus 1 Virus 2AUT 12 Prakash and Faloutsos 2012 55
    56. 56. Question: What happens in the end?Number of green: virus 1Infections red: virus 2 Footprint @ Steady State Footprint @ Steady State = ? ASSUME: Virus 1 is stronger than Virus AUT 12 Prakash and Faloutsos 2012 56
    57. 57. Question: What happens in the end? Footprint @ Steady StateNumber of green: virus 1 Footprint @ Steady StateInfections red: virus 2 Strength Strength ?? = 2 Strength Strength ASSUME: Virus 1 is stronger than Virus AUT 12 Prakash and Faloutsos 2012 57
    58. 58. Answer: Winner-Takes-AllNumber of green: virus 1Infections red: virus 2 ASSUME: Virus 1 is stronger than Virus AUT 12 Prakash and Faloutsos 2012 58
    59. 59. Our Result: Winner-Takes-All Given our model, and any graph, the weaker virus always dies-out completely1. The stronger survives only if it is above threshold2. Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus 2)4. Strength(Virus) = λ β / δ  same as before!In Prakash+ WWW 2012 AUT 12 Prakash and Faloutsos 2012 59
    60. 60. Real Examples[Google Search Trends data] Reddit v Digg Blu-Ray v HD-DVDAUT 12 Prakash and Faloutsos 2012 60
    61. 61. Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms)AUT 12 Prakash and Faloutsos 2012 61
    62. 62. Full Static ImmunizationGiven: a graph A, virus prop. model and budget k;Find: k ‘best’ nodes for immunization (removal). ? ? k=2 ? ? AUT 12 Prakash and Faloutsos 2012 62
    63. 63. Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional ImmunizationAUT 12 Prakash and Faloutsos 2012 63
    64. 64. 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’?AUT 12 Prakash and Faloutsos 2012 64
    65. 65. Proposed vulnerability measure λ λ is the epidemic threshold “Safe” “Vulnerable” “Deadly” Increasing λAUT 12 Increasing vulnerability Prakash and Faloutsos 2012 65
    66. 66. A1: “Eigen-Drop”: an ideal shield value Eigen-Drop(S) Δ λ = λ - λs 9 9 11 9 10 Δ 10 1 1 4 4 8 8 2 2 7 3 7 3 5 5 6 6 Original Graph Without {2, 6} 66AUT 12 Prakash and Faloutsos 2012
    67. 67. (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!AUT 12 Prakash and Faloutsos 2012 67
    68. 68. 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(nk2+m) n = # nodes; m = # edgesIn Tong, Prakash+ ICDM 2010 AUT 12 Prakash and Faloutsos 2012 68
    69. 69. Experiment: ImmunizationLog(fraction of qualityinfectednodes) PageRank Betweeness (shortest path) DegreeLower Acquaintance is NetShield Eigs (=HITS)better Time AUT 12 Prakash and Faloutsos 2012 69
    70. 70. Outline• Motivation• Epidemics: what happens? (Theory)• Action: Who to immunize? (Algorithms) – Full Immunization (Static Graphs) – Fractional ImmunizationAUT 12 Prakash and Faloutsos 2012 70
    71. 71. Fractional Immunization of NetworksB. Aditya Prakash, Lada Adamic, TheodoreIwashyna (M.D.), Hanghang Tong, ChristosFaloutsosUnder reviewAUT 12 Prakash and Faloutsos 2012 71
    72. 72. Fractional Asymmetric Immunization Drug-resistant Bacteria (like XDR-TB) Hospital Another HospitalAUT 12 Prakash and Faloutsos 2012 72
    73. 73. Fractional Asymmetric Immunization Drug-resistant Bacteria (like XDR-TB) Hospital Another HospitalAUT 12 Prakash and Faloutsos 2012 73
    74. 74. Fractional Asymmetric Immunization Problem: Given k units of disinfectant, how to distribute them to maximize hospitals saved? Hospital Another HospitalAUT 12 Prakash and Faloutsos 2012 74
    75. 75. Our Algorithm “SMART- ALLOC” ~6x [US-MEDICARE NETWORK 2005] fewer! • Each circle is a hospital, ~3000 hospitals • More than 30,000 patients transferredCURRENT PRACTICE SMART-ALLOC 75AUT 12 Prakash and Faloutsos 2012
    76. 76. Running TimeWall-Clock > 1 weekTime ≈ > 30,000x speed-up! Lower is 14 secs better Simulations SMART-ALLOC AUT 12 Prakash and Faloutsos 2012 76
    77. 77. Lower Experimentsisbetter PENN-NETWORK SECOND-LIFE ~5 x ~2.5 xAUT 12 K = 200 Prakash and Faloutsos 2012 K = 2000 77
    78. 78. AcknowledgementsFundingAUT 12 Prakash and Faloutsos 2012 78
    79. 79. References• 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.)• 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• 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• Winner-takes-all: Competing Viruses or Ideas on fair-play networks (B. Aditya Prakash, Alex Beutel, Roni Rosenfeld, Christos Faloutsos) – In WWW 2012, Lyon• On the Vulnerability of Large Graphs (Hanghang Tong, B. Aditya Prakash, Tina Eliassi- Rad and Christos Faloutsos) – In IEEE ICDM 2010, Sydney, Australia• Fractional Immunization of Networks (B. Aditya Prakash, Lada Adamic, Theodore Iwashyna, Hanghang Tong, Christos Faloutsos) - Under Submission• Rise and Fall Patterns of Information Diffusion: Model and Implications (Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos) - Under Submission http://www.cs.cmu.edu/~badityap/AUT 12 Prakash and Faloutsos 2012 79
    80. 80. Propagation on Large NetworksB. Aditya PrakashChristos FaloutsosAnalysis Policy/Action Data AUT 12 Prakash and Faloutsos 2012 80

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