CS 6604: Data Mining Large Networks and Time-series. B. Aditya Prakash Lecture #8: Epidemics: Thresholds

Transcription

1 CS 6604: Data Mining Large Networks and Time-series B. Aditya Prakash Lecture #8: Epidemics: Thresholds

2 A fundamental Strong Virus Epidemic? 2

3 example graph) Weak Virus Epidemic? 3

4 Problem Statement # Infected above (epidemic) below Find, a condi4on under which virus will die out exponen4ally quickly regardless of ini4al infec4on condi4on Separate the regimes? 4

5 Threshold version) Problem Statement Given: Graph G, and Virus specs (apack prob. etc.) Find: A condiuon for virus exuncuon/invasion 5

6 Threshold: Why important? AcceleraUng simulauons ForecasUng ( What-if scenarios) Design of contagion and/or topology A great handle to manipulate the spreading ImmunizaUon Maximize collaborauon.. 6

7 Outline Q: What is the epidemic threshold? Background Result and IntuiUon (StaUc Graphs) Proof Ideas (StaUc Graphs) Bonus: Dynamic Graphs 7

8 SIR model: life immunity (mumps) Each node in the graph is in one of three states SuscepUble (i.e. healthy) Infected Removed (i.e. can t get infected again) Prob. β Prob. δ t = 1 t = 2 t = 3 8

9 Related Work q q q q q q q R. M. Anderson and R. M. May. InfecUous Diseases of Humans. Oxford University Press, A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex Networks. Cambridge University Press, F. M. Bass. A new product growth for model consumer durables. Management Science, 15(5): , D. ChakrabarU, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos. Epidemic thresholds in real networks. ACM TISSEC, 10(4), D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, A. Ganesh, L. Massoulie, and D. Towsley. The effect of network topology in spread of epidemics. IEEE INFOCOM, Y. Hayashi, M. Minoura, and J. Matsukubo. Recoverable prevalence in growing scale-free networks and the effecuve immunizauon. arxiv:cond-at/ v2, Aug q H. W. Hethcote. The mathemaucs of infecuous diseases. SIAM Review, 42, q H. W. Hethcote and J. A. Yorke. Gonorrhea transmission dynamics and control. Springer Lecture Notes in BiomathemaUcs, 46, q q q q q q J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses. IEEE Computer Society Symposium on Research in Security and Privacy, J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. IEEE Computer Society Symposium on Research in Security and Privacy, R. Pastor-Santorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review LePers 86, 14, All are about either: Structured topologies (cliques, block-diagonals, hierarchies, random) Specific virus propaga@on models Sta@c graphs 9

10 Outline Q: What is the epidemic threshold? Background Result and Graphs) Proof Ideas (StaUc Graphs) Bonus: Dynamic Graphs 10

11 How should the answer look like? Answer should depend on: Graph Virus PropagaUon Model (VPM) But how?? β + δ d avg Graph average degree? max. degree? diameter? VPM which parameters? How to combine linear? quadrauc? exponenual? diameter? ( avg max 2 2 β d avg δd ) / d?.. 11

12 Graphs: Main Result [Prakash+, 2011] Informally, For, Ø any arbitrary topology (adjacency matrix A) Ø 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 C VPM, determined by the virus propagation model λ C VPM No epidemic if λ * C VPM < 1 12

13 Our thresholds for some models s = effec4ve strength s < 1 : below threshold Models Effec@ve Strength (s) Threshold (@pping point) SIS, SIR, SIRS, SEIR s = λ. SIV, SEIV s = λ. SI 1I2V1 V2 (H.I.V.) s = λ. β δ βγ δ β1v v2 ( γ +θ ) 2 + β ε 2 ( ε + v ) 1 s = 1 13

14 Our result: for λ Official Let A be the adjacency matrix. Then λ is the root with the largest magnitude of the characteristic polynomial of A [det(a xi)]. Un-official Intui@on J λ ~ # paths in the graph k A λ k. u u Doesn t give much intuiuon! A k(i, j) = # of paths i à j of length k 14

15 Largest Eigenvalue (λ) beper connecuvity higher λ λ 2 λ = N λ = N-1 λ 2 λ= λ= 999 N = 1000 N nodes 15

16 Examples: SIR (mumps) of Footprint Strength (a) Infection profile (b) Take-off plot PORTLAND graph 31 million CS 6604:DM links, Large Networks 6 million & Time-Series nodes 16

17 Examples: SIRS (pertusis) of Footprint Strength (a) Infection profile (b) Take-off plot PORTLAND graph 31 million CS 6604:DM links, Large Networks 6 million & Time-Series nodes 17

18 Outline Q: What is the epidemic threshold? Background Result and IntuiUon (StaUc Graphs) Proof Ideas Graphs) Bonus: Dynamic Graphs 18

19 Proof Sketch General VPM structure Model-based λ * C VPM < 1 Topology and stability Graph-based 19

20 Some trivia Ø first person in the US idenufied as a healthy carrier of the pathogen associated with typhoid fever. Ø infected some 53 people, over the course of her career as a cook! Ø forcibly quaranuned by public health authoriues 20

21 Two Infected States? SICR: with a carrier AsymtomaUc SymptomaUc Sneezing I1 I2 21

22 Ingredient 1: Our generalized model Endogenous Infected Exogenous Endogenous Vigilant 22

23 Models and more models Model SIR SIS SIRS SEIR.. SICR MSIR SIV SI 1I2V1 V2. Used for Mumps Flu Pertussis Chicken-pox Tuberculosis Measles Sensor Stability H.I.V. 23

24 Our generalized model Endogenous Infected Vigilant 24

25 Special case: SIR Infected Vigilant 25

26 Special case: H.I.V. SI I V V Non-terminal Terminal MulUple InfecUous, Vigilant states 26

27 Ingredient 2: NLDS + Stability View as a NLDS discrete Ume non-linear dynamical system (NLDS) size mn x Probability vector Specifies the state of the system at Ume t size N (number of nodes in the graph) S I V 27

28 Ingredient 2: NLDS + Stability View as a NLDS discrete Ume non-linear dynamical system (NLDS) size mn x 1... Non-linear func@on Explicitly gives the evoluuon of system

29 Ingredient 2: NLDS + Stability View as a NLDS discrete Ume non-linear dynamical system (NLDS) Threshold à Stability of NLDS 29

30 Special case: SIR S size 3N x 1 I R S I R = probability that node i is not apacked by any of its infecuous neighbors NLDS 30

31 Fixed Point State when no node is infected Q: Is it stable? 31

32 Stability for SIR Stable Unstable under threshold above threshold 32

33 See paper for full proof General VPM structure Model-based λ * C VPM < 1 Topology and stability Graph-based 33

34 Outline Q: What is the epidemic threshold? Background Result and IntuiUon (StaUc Graphs) Proof Ideas (StaUc Graphs) Bonus: Dynamic Graphs 34

35 Dynamic Graphs: Epidemic? DAY (e.g., work) Alternating behaviors adjacency matrix

36 Dynamic Graphs: Epidemic? NIGHT (e.g., home) Alternating behaviors adjacency matrix

37 Model Healthy SIS model recovery rate δ infecuon rate β N1 Infected Prob. β X N2 N3 Prob. δ Set of T arbitrary graphs day N night N, weekend.. N N 37

38 Obvious result No epidemic if λ max β δ <1 #inf. This looks OK BUT Too pessimisuc! Ume 38

39 Main result: Dynamic Graphs Threshold [Prakash+, 2010] Informally, NO epidemic if eig (S) = < 1 Single number! Largest eigenvalue of The system matrix S 39

40 NO epidemic if eig (S) = < 1 S = cure rate Prob. β N1 Infected Prob. δ X Healthy N2 infec@on rate N3 N day adjacency matrix night.. N 40

41 log(frac4on infected) ABOVE MIT Reality Mining ABOVE AT AT BELOW BELOW Time 41

42 Footprint (# steady state ) Take-off plots Synthe@c MIT Reality Our threshold EPIDEMIC Our threshold EPIDEMIC NO EPIDEMIC NO EPIDEMIC (log scale) 42