Impact of Clustering on Epidemics in Random Networks

Transcription

1 Impact of Clustering on Epidemics in Random Networks Joint work with Marc Lelarge INRIA-ENS 8 March 2012 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

2 Outline 1 Introduction : Social Networks and Epidemics 2 Random Graph Model 3 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

3 Introduction : Social Networks and Epidemics Outline 1 Introduction : Social Networks and Epidemics 2 Random Graph Model 3 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

4 Introduction : Social Networks and Epidemics Game-theoretic contagion model on a given graph G = (V, E), with parameter q (0, 1/2) : Two possible choices : ( susceptible) or ( infected) Initially : all use, except one who uses Possible switch, but no switch Situation Payoff (for both users) q 1 q > q 0 Total payoff = sum of payoffs from all your neighbors Switch from to Neighbors using Skype Neighbors > q. Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

5 Introduction : Social Networks and Epidemics Infinite deterministic graph G = (V, E) Parameter q varies : q small CASCADE q higher NO cascade More precisely : q 1 q 2, cascade for q 1 cascade for q 2 Contagion threshold q (G) c := sup { q CASCADE in G for parameter q } CASCADE threshold q (G) c NO cascade q Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

6 Introduction : Social Networks and Epidemics Switch from to Neighbors using Skype Neighbors > q Example : G = d-regular tree q 1/d NO cascade q < 1/d CASCADE q (G) c = 1/d Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

7 Introduction : Social Networks and Epidemics Switch from to Neighbors using Skype Neighbors > q Example : G = d-regular tree q 1/d NO cascade q < 1/d CASCADE q (G) c = 1/d Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

8 Introduction : Social Networks and Epidemics Switch from to Neighbors using Skype Neighbors > q Example : G = d-regular tree q 1/d NO cascade q < 1/d CASCADE q (G) c = 1/d Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

9 Introduction : Social Networks and Epidemics Switch from to Neighbors using Skype Neighbors > q Example : G = d-regular tree q 1/d NO cascade q < 1/d CASCADE q (G) c = 1/d Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

10 Introduction : Social Networks and Epidemics Define a model of finite random graphs (whose size tends to infinity) having (asymptotically) the observed properties : scale-free networks power law degree distribution i.e. there exists τ > 0 such that, for all k 0, p k k τ (small number of nodes having a large number of edges) networks with clustering ( The friends of my friends are my friends, Newman, 03) tractable Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

11 Introduction : Social Networks and Epidemics Epidemic models on finite random graphs : Final nb of infected nodes negligeable or not / population size? G n = random graph of size n S n = final size of the epidemic in G n CASCADE if S n = Θ p (n), n NO cascade if S n = o p (n). n Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

12 Introduction : Social Networks and Epidemics Epidemic models on finite random graphs : Final nb of infected nodes negligeable or not / population size? CONTAGION MODEL DIFFUSION MODEL Ref. Morris, Watts Bond percolation Para- A vertex is infected π = probability that an edge -meter fraction of infected neighbors > q transmits the epidemic 1 3 q > q π f 1 π f Thm CASCADE threshold q c NO cascade q NO cascade threshold π c CASCADE π Effect of clustering on these thresholds and on the cascade size Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

13 Random Graph Model Outline 1 Introduction : Social Networks and Epidemics 2 Random Graph Model 3 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

14 Random Graph Model (i) Start from a uniform graph with given vertex degrees (ii) Add clustering (i) Original graph (with given vertex degrees) : n N, d = (d i ) n 1 sequence of non-negative integers (s.t. a graph with n vertices and degree sequence d). G (n, d) = uniform random graph (among the graphs with n vertices and degree sequence d). d 1 = 3 d 2 = 2 d n = 5 Ref. : (Lelarge, 11) for the study of contagion and diffusion models on graphs with given vertex degrees Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

15 Random Graph Model (i) Start from a uniform graph with given vertex degrees (ii) Add clustering (i) Original graph (with given vertex degrees) : n N, d = (d i ) n 1 sequence of non-negative integers (s.t. a graph with n vertices and degree sequence d). G (n, d) = uniform random graph (among the graphs with n vertices and degree sequence d). Condition : Assume there exists a probability distribution p = (p r ) r=0 such that : (i) {i : d i = r}/n p r as n, for every r 0 (ii) λ := r rpr (0; ) (iii) i d 3 i = O(n) Ref. : (Lelarge, 11) for the study of contagion and diffusion models on graphs with given vertex degrees Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

16 Random Graph Model (ii) Clustering Coefficient of G = (V, E) : C (G) := probability that two vertices share an edge together, C (G) := knowing that they have a common neighbor C (G) = 3 nb of triangles nb of connected triples = v P v v N v P v := nb of pairs of neighbors of v sharing an edge together, N v := nb of pairs of neighbors of v : N v = d v (d v 1)/2. Example : N v = 3 P v = 0 v P v = 2 v P v = 3 v Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

17 Random Graph Model Idea : Replace a vertex of degree r in G (n, d) by a clique of size r : Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

18 Random Graph Model Idea : Replace a vertex of degree r in G (n, d) by a clique of size r. Adding cliques randomly : Let γ [0, 1]. Each vertex is replaced by a clique with probability γ (independently for all vertices). G (n, d, γ) = resulting random graph (with additional cliques) Similar model : (Trapman, 07), (Gleeson, 09) Particular cases : γ = 0 G (n, d, γ) = G (n, d), γ = 1 all vertices in G (n, d) have been replaced by cliques. New asymptotic degree distribution p = ( p k ) k 0 Asymptotic clustering coefficient C > 0 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

19 Outline 1 Introduction : Social Networks and Epidemics 2 Random Graph Model 3 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

20 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q Heuristically... The random graph G (n, d) converges locally to a random tree such that : P (r 1 children) = rp r /λ Infected nodes = those with degree < 1/q q = 1 4 Infinite tree (of infected nodes) (r 1) rp r λ > 1 r<1/q Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

21 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q Heuristically... The random graph G (n, d) converges locally to a random tree such that : P (r 1 children) = rp r /λ Infected nodes = those with degree < 1/q q = 1 4 Infinite tree (of infected nodes) (r 1) rp r λ > 1 r<1/q Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

22 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q Heuristically... The random graph G (n, d) converges locally to a random tree such that : P (r 1 children) = rp r /λ Infected nodes = those with degree < 1/q q = 1 4 Infinite tree (of infected nodes) (r 1) rp r λ > 1 r<1/q Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

23 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q Heuristically... The random graph G (n, d) converges locally to a random tree such that : P (r 1 children) = rp r /λ Infected nodes = those with degree < 1/q q = 1 4 Infinite tree (of infected nodes) (r 1) rp r λ > 1 r<1/q Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

24 { q c := q c (p) = sup q : } r<1/q (r 1)rp r λ > 1 Fixed q, P (n) = set of pivotal players in G (n, d, γ) : G 0 = induced subgraph with vertices of degree < 1/q Pivotal players = vertices in the largest connected component of G 0 Theorem (CONTAGION THRESHOLD) q < q c : P (n) = Θ p (n) Each pivotal player can trigger a global cascade. q > q c : the size of the epidemic generated by a vertex u (chosen uniformly at random) is negligeable : o p (n). Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

25 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

26 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

27 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

28 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

29 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

30 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

31 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

32 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

33 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

34 Contagion model with parameter q : At the beginning, one infected vertex (= the seed of the epidemic) At each step, each vertex becomes infected if : proportion of its infected neighbors > q = 1 4 INITIAL GRAPH GRAPH WITH CLIQUES = Clustering decreases the cascade size. Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

35 Contagion Threshold (q c ) vs Mean Degree Graphs with the SAME asymptotic degree distribution : p k k τ e k/50 Graph with clustering (cliques) Graph with no clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

36 Effect of Clustering on the Contagion Threshold Asymptotic degree distribution : p k k τ e k/50 Mean degree λ 1.65 No Cascade Cascade Graph with maximal clustering coefficient Graph with no clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

37 Effect of Clustering on the Contagion Threshold Asymptotic degree distribution : p k k τ e k/50 Mean degree λ 46 No Cascade Cascade Graph with maximal clustering coefficient Graph with no clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

38 Effect of Clustering on the Contagion Threshold Asymptotic degree distribution : p k k τ e k/50 Mean degree λ 3.22 No Cascade Cascade Graph with maximal clustering coefficient Graph with no clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

39 Effect of Clustering on the Cascade Size Fraction of infected neighbors needed to become infected : q = 0.15 (fixed) p r = 0.2r+0.8 e λ λ r 0.2λ+0.8 r! C = 0 Pivotal players in the graph with no clustering Cascade size in the graph with no clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

40 Effect of Clustering on the Cascade Size Fraction of infected neighbors needed to become infected : q = 0.15 (fixed) p r = 0.2r+0.8 e λ λ r 0.2λ+0.8 r! C = 0.2λ 0.2λ+1.2 > 0 Pivotal players in the graph with positive clustering Cascade size in the graph with positive clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

41 Effect of Clustering on the Cascade Size Fraction of infected neighbors needed to become infected : q = 0.15 (fixed) Pivotal players in the graph with no clustering Cascade size in the graph with no clustering Pivotal players in the graph with positive clustering Cascade size in the graph with positive clustering Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

42 Effect of Clustering on the Cascade Size Asymptotic degree distribution : p k k τ e k/50 Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

43 Conclusion Model of random graphs with a given degree distribution, and a tunable clustering coefficient Effect of clustering on the contagion model : Clustering decreases the contagion threshold for low values of the mean degree, while the opposite happens in the high values regime Clustering decreases the cascade size (when a cascade is possible) For the following questions, see our paper on arxiv : : Effect of clustering on the diffusion model (bond percolation) Positive proportion of the population initially infected Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

44 Conclusion Model of random graphs with a given degree distribution, and a tunable clustering coefficient Effect of clustering on the contagion model : Clustering decreases the contagion threshold for low values of the mean degree, while the opposite happens in the high values regime Clustering decreases the cascade size (when a cascade is possible) For the following questions, see our paper on arxiv : : Effect of clustering on the diffusion model (bond percolation) Positive proportion of the population initially infected Thanks for your attention! If you liked the presentation, I am looking for a post-doc position for September Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19

45 References T. Britton, M. Deijfen, A. N. Lagerås, and M. Lindholm. Epidemics on random graphs with tunable clustering. J. Appl. Probab., 45(3) : , A. Hackett, S. Melnik, and J. P. Gleeson. Cascades on a class of clustered random networks. Physical Review E, 83, M. Lelarge. Diffusion and cascading behavior in random networks. under revision for Games and Economic Behavior, arxiv/ , M. E. J. Newman. Properties of highly clustered networks. Phys. Rev. E, 68(2) :026121, Aug P. Trapman. On analytical approaches to epidemics on networks. Theoretical Population Biology, 71(2) : , D. J. Watts and S. H. Strogatz. Collective dynamics of small-world networks. Nature, 393(6684) : , June [1, 3, 6, 4, 5, 2] Coupechoux - Lelarge (INRIA-ENS) Epidemics in Random Networks 8 March / 19