Epidemics and information spreading

Transcription

1 Epidemics and information spreading Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network Analysis MAGoLEGO course Leonid E. Zhukov (HSE) Lecture / 36

2 Lecture outline 1 Epidemics SI model SIS model SIR model 2 Information spread 3 Propagation trees Leonid E. Zhukov (HSE) Lecture / 36

3 Flu contagion Infected - red, friends of infected - yellow N. Christakis, J. Fowler, 2010 Leonid E. Zhukov (HSE) Lecture / 36

4 Global contagion Simulated epidemic: gray lines - passenger flow, red symbols epidemics location D. Brockmann, D. Helbing, 2013 Leonid E. Zhukov (HSE) Lecture / 36

5 Network epidemic model Given a network G of potential contacts Three states model: susceptible, infected, recovered Probabilistic model (state of a node): s i (t) - probability that at t node i is susceptible x i (t) - probability that at t node i is infected r i (t) - probability that at t node i is recovered β - infection rate (probably to get infected on a contact in time δt) γ - recovery rate (probability to recover in a unit time δt) connected component - all nodes reachable network is undirected (matrix A is symmetric) if graph complete - fully mixing model Based upon models from mathematical epidemiology, W.O. Kermack and McKendrick, 1927 Leonid E. Zhukov (HSE) Lecture / 36

6 Probabilistic model Two processes: Node infection: Node recovery: P inf βs i (t) j N (i) x j (t)δt P rec = γx i (t)δt Leonid E. Zhukov (HSE) Lecture / 36

7 SI model SI Model S I Probabilities that node i: s i (t) - susceptible, x i (t) -infected at t x i (t) + s i (t) = 1 β - infection rate, probability to get infected in a unit time x i (t + δt) = x i (t) + βs i A ij x j δt infection equations j dx i (t) dt = βs i (t) j A ij x j (t) x i (t) + s i (t) = 1 Leonid E. Zhukov (HSE) Lecture / 36

8 SI simulation 1 Every node at any time step is in one state {S, I } 2 Initialize c nodes in state I 3 On each time step each I node has a probability β to infect its nearest neighbors (NN), S I Model dynamics: I + S β 2I Leonid E. Zhukov (HSE) Lecture / 36

9 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

10 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

11 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

12 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

13 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

14 SI model β = 0.5 Leonid E. Zhukov (HSE) Lecture / 36

15 SI model Leonid E. Zhukov (HSE) Lecture / 36

16 SI model 1 growth rate of infections depends on λ 1 2 All nodes in connected component get infected t x i (t) 1 3 probability of infection of nodes depends on v 1, i.e v 1i image from M. Newman, 2010 Leonid E. Zhukov (HSE) Lecture / 36

17 SIS model SIS Model S I S Probabilities that node i: s i (t) - susceptible, x i (t) -infected at t x i (t) + s i (t) = 1 β - infection rate, γ - recovery rate infection equations: dx i (t) dt = βs i (t) j A ij x j (t) γx i x i (t) + s i (t) = 1 Model dynamics: { I + S I β γ 2I S Leonid E. Zhukov (HSE) Lecture / 36

18 SIS simulation 1 Every node at any time step is in one state {S, I } 2 Initialize c nodes in state I 3 Each node stays infected τ γ = 0 τe τγ dτ = 1/γ time steps 4 On each time step each I node has a probability β to infect its nearest neighbors (NN), S I 5 After τ γ time steps node recovers, I S Model dynamics: { I + S I β γ 2I S Leonid E. Zhukov (HSE) Lecture / 36

19 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

20 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

21 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

22 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

23 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

24 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

25 SIS model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

26 SIS model Leonid E. Zhukov (HSE) Lecture / 36

27 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

28 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

29 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

30 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

31 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

32 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

33 SIS model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

34 SIS model Leonid E. Zhukov (HSE) Lecture / 36

35 In SIS model: R 0 = 1 λ 1, Av 1 = λ 1 v 1 Leonid E. Zhukov (HSE) Lecture / 36 SIS model Epidemic threshold R 0 : if β γ < R 0 - infection dies over time if β γ > R 0 - infection survives and becomes epidemic

36 SIR model SIR Model S I R probabilities s i (t) -susceptable, x i (t) - infected, r i (t) - recovered s i (t) + x i (t) + r i (t) = 1 β - infection rate, γ - recovery rate Infection equation: dx i dt = βs i A ij x j γx i j dr i dt = γx i x i (t) + s i (t) + r i (t) = 1 Leonid E. Zhukov (HSE) Lecture / 36

37 SIR simulation 1 Every node at any time step is in one state {S, I, R} 2 Initialize c nodes in state I 3 Each node stays infected τ γ = 1/γ time steps 4 On each time step each I node has a prabability β to infect its nearest neighbours (NN), S I 5 After τ γ time steps node recovers, I R 6 Nodes R do not participate in further infection propagation Model dynamics: { I + S I β γ 2I R Leonid E. Zhukov (HSE) Lecture / 36

38 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

39 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

40 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

41 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

42 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

43 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

44 SIR model β = 0.5, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

45 SIR model Leonid E. Zhukov (HSE) Lecture / 36

46 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

47 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

48 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

49 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

50 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

51 SIR model β = 0.2, τ = 2 Leonid E. Zhukov (HSE) Lecture / 36

52 SIR model Leonid E. Zhukov (HSE) Lecture / 36

53 SIR model Epidemic threshold R 0 : β γ > R 0 - infection survives and becomes epidemic Leonid E. Zhukov (HSE) Lecture / 36

54 SIR model Epidemic threshold R 0 : β γ < R 0 - infection dies over time Leonid E. Zhukov (HSE) Lecture / 36

55 5 Networks, SIR Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free image from Keeling et al, 2005 Leonid E. Zhukov (HSE) Lecture / 36

56 5 Networks, SIR Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free Keeling et al, 2005 Leonid E. Zhukov (HSE) Lecture / 36

57 Effective distance J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013 Leonid E. Zhukov (HSE) Lecture / 36

58 Social contagion Social contagion phenomena refer to various processes that depend on the individual propensity to adopt and diffuse knowledge, ideas, information. Similar to epidemiological models: - susceptible - an individual who has not learned new information - infected - the spreader of the information - recovered - aware of information, but no longer spreading it Two main questions: - if the rumor reaches high number of individuals - rate of infection spread Leonid E. Zhukov (HSE) Lecture / 36

59 Branching process Simple contagion model: - each infected person meets k new people - independently transmits infection with probability p image from David Easley, Jon Kleinberg, 2010 Leonid E. Zhukov (HSE) Lecture / 36

60 Branching process Galton-Watson branching random process, R 0 = pk if R 0 = 1, the mean of number of infected nodes does not change if R 0 > 1, the mean grows geometrically as R0 n if R 0 < 1, the mean shrinks geometrically as R0 n R 0 = pk = 1 - point of phase transition Leonid E. Zhukov (HSE) Lecture / 36

61 Propagation tree Rumors/news/infection spreads over the edges: propagation tree (graph) Sometimes called cascade Leonid E. Zhukov (HSE) Lecture / 36

62 Online newsletter experiment Online newsletter + reward for recommending it 7,000 seed, 30,000 total recipients, around 7,000 cascades, size form 2 nodes to 146 nodes, max depth 8 steps on average k = 2.96, infection probability p = Suprisingly: no loops, triangles, closed paths Iribarren et.al, 2009 Leonid E. Zhukov (HSE) Lecture / 36

63 Mem diffusion Mem diffusion on Twitter L. Weng et.al, 2012 Leonid E. Zhukov (HSE) Lecture / 36

64 References Epidemic outbreaks in complex heterogeneous networks. Y. Moreno, R. Pastor-Satorras, and A. Vespignani. Eur. Phys. J. B 26, , Networks and Epidemics Models. Matt. J. Keeling and Ken.T.D. Eames, J. R. Soc. Interfac, 2, , 2005 Simulations of infections diseases on networks. G. Witten and G. Poulter. Computers in Biology and Medicine, Vol 37, No. 2, pp , 2007 Dynamical processes on complex networks. A. Barrat, M. Barthelemy, A. Vespignani Eds., Cambridge University Press 2008 Dynamics of rumor spreading in complex networks. Y. Moreno, M. Nekovee, A. Pacheco, Phys. Rev. E 69, , 2004 Theory of rumor spreading in complex social networks. M. Nekovee, Y. Moreno, G. Biaconi, M. Marsili, Physica A 374, pp , Impact of Human Activity Patterns on the Dynamics of Information Diffusion. J.L. Iribarren, E. Moro, Phys.Rev.Lett. 103, , Leonid E. Zhukov (HSE) Lecture / 36