Susceptible-Infective-Removed Epidemics and Erdős-Rényi random

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1 Susceptible-Infective-Removed Epidemics and Erdős-Rényi random graphs MSR-Inria Joint Centre October 13, 2015

2 SIR epidemics: the Reed-Frost model Individuals i [n] when infected, attempt to infect all neighbors independently with probability p in subsequent time slot and then die Focus on complete graph (everyone neighbor of everyone) Associated model: Erdős-Rényi random graph G(n, p): undirected graph on node set [n]. Edge (i, j) present iff ξ ij = 1 where {ξ ij } i<j : i.i.d., Bernoulli (p)

3 SIR epidemics: the Reed-Frost model Individuals i [n] when infected, attempt to infect all neighbors independently with probability p in subsequent time slot and then die Focus on complete graph (everyone neighbor of everyone) Associated model: Erdős-Rényi random graph G(n, p): undirected graph on node set [n]. Edge (i, j) present iff ξ ij = 1 where {ξ ij } i<j : i.i.d., Bernoulli (p) From random graph to epidemic process: use ξ ij to determine if when the first of i and j gets infected, it infects the other

4 SIR epidemics: the Reed-Frost model Individuals i [n] when infected, attempt to infect all neighbors independently with probability p in subsequent time slot and then die Focus on complete graph (everyone neighbor of everyone) Associated model: Erdős-Rényi random graph G(n, p): undirected graph on node set [n]. Edge (i, j) present iff ξ ij = 1 where {ξ ij } i<j : i.i.d., Bernoulli (p) From random graph to epidemic process: use ξ ij to determine if when the first of i and j gets infected, it infects the other For initial set X 0 of infective nodes at time 0, i infected at time t iff d G (X 0, i) = t Set of nodes eventually infected: i X0 Γ(i) where Γ(i): graph s connected component including i

5 Outline Seminal results by Erdős and Rényi ( ) First phase transition: emergence of giant component [Tool: branching processes & Chernoff s inequality] Second phase transition: emergence of connectivity [Tools: 1st and 2nd moment methods; Poisson approximation]

6 Emergence of giant component Analysis of graph s connected components: let C(i): size of i-th largest connected component (in number of nodes) in G(n, p) Theorem Let p = λ/n for fixed λ > 0 Sub-critical case (λ < 1): there exists f (λ) such that lim P(C(1) f (λ) log(n)) = 1 n Super-critical case (λ > 1): there exists g(λ) such that for all δ > 0, lim P( C(1) (1 p ext ) δ, C(2) g(λ) log(n)) = 1, n n where p ext : extinction probability of Poisson (λ) branching process, i.e. smallest root of x = e λ(x 1) in [0, 1]

7 Interpretation Sub-critical regime: Only logarithmically sized components i.e. no global outbreak Super-critical regime: with probability 1 p ext, epidemics started from randomly selected node reaches n[1 p ext + o(1)] others, i.e. macroscopic outbreak Note: only one giant component, others still logarithmic

8 Sub-critical regime Exploration of connected component Γ(i 0 ): initialized with active set A 0 = {i 0 } and killed set B 0 = At time t pick j t A t 1, kill it and activate its neighbours not yet activated (set D t ) A t = A t 1 \ {j t } D t, B t = B t 1 {j t } Notation: A t = A t, D t = D t A t = 1 t + D D t Conditionally on F t 1 = σ(a 0,..., A t 1 ), D t Bin(p, n 1 D 0 D t 1 ) Size C of connected component: C = inf{t > 0 : A t = 0}

9 Sub-critical regime, continued Processes {A t }, {D t } can be extended after end of component s exploration Upper bound: P(C > k) = P(A 1,..., A k > 0) P(A k > 0) Chernoff s bounding technique: P(A k > 0) e kh(1) where h(x) = λh 1 (x/λ), h 1 (x) = x log(x) x + 1: Chernoff s exponent for Poisson (λ) random variable Union bound allows to conclude

10 Super-critical regime λ > 1 Lemma For any k > 0, d 1,..., d k N k, lim n P(D1 k = d 1 k) = k λds s=1 e λ d, s! hence lim n P(C k) = P(Z k) where Z: total population of Poisson (λ) branching process Lemma For any k 1, A k + k 1 Bin(1 (1 p) k, n 1), hence P(C = k) P(Bin(1 (1 p) k, n 1) = k 1) Corollary For any fixed ɛ, δ > 0, there exists A > 0 such that P(C A) p ext ɛ, P( C/n (1 p ext ) δ) 1 p ext ɛ

11 Super-critical regime, continued Fix ɛ, δ > 0. Call connected component small if C A, gigantic if C/n (1 p ext ) δ, failed otherwise. Repeatedly extract connected component until either failure or giant component found Probability of finding giant, i.e. success, in at most M steps: at least M 1 (p ext ɛ) m [1 p ext ɛ] = 1 p ext ɛ 1 p ext + ɛ [1 (p ext ɛ) M ] m=0 Hence probability of success 1 O(ɛ) for M Ω(log(1/ɛ)) Given success, remaining graph: G(n, p) with n n n(1 p ext δ) = n(p ext + δ) a sub-critical Erdős-Rényi graph, as λp ext < 1 (see notes on branching processes)

12 Connectivity By previous result: for fixed λ > 1, giant component of size n(1 p ext ) For fixed λ, graph disconnected Under what regime is graph connected? Theorem For fixed c R, assume np = log(n) + c. Then lim n P(G(n, p) connected) = e e c Corollary If np log(n) +, then lim n P(G(n, p) connected) = 1 If np log(n), then lim n P(G(n, p) connected) = 0

13 Proof strategy Show that number of isolated nodes (i.e. nodes of degree 0) admits asymptotically Poisson (e c ) distribution [Poisson approximation method], hence lim n P(A) = e e c where A = {no isolated vertices in G(n, p)} Show that lim n P(B) = 0 where B = { connected component of size k {2,..., n/2}} Use bounds P(A) P(B) P(G(n, p) connected) = P(A B) P(A)

14 Basic tools: the first and second moment methods Let Z u, u V be indicators of events and X = u V Z u. First moment method: P( u V : Z u = 1) u V E(Z u) = E(X ), hence with high probability none of these events occurs if lim n E(X ) = 0. Application: with high probability no isolated node in G(n, p) if lim n [np log(n)] = +. Second moment method: P( u V, Z u = 0) = P(X = 0) Var(X ) E(X ) 2. Hence if Var(X ) = o(e(x ) 2 ), then with high probability some event occurs. Application: with high probability there is some isolated node in G(n, p) if lim n [np log(n)] =.

15 Variation distance Definition Variation distance between two probability measures µ, ν on (Ω, F): d var (µ, ν) = 2 sup A F µ(a) ν(a) Alternative characterization: if µ, ν admit densities dµ respect to measure π (e.g., π = µ + ν) then d var (µ, ν) = Ω dµ dπ dν dπ dπ dπ, dν dπ with In particular for Ω = N and π = δ n, d var (µ, ν) = µ n ν n n N n N Definition {µ (n) } n N converges in variation to µ iff lim n d var (µ (n), µ) = 0 A strong form of convergence (implies convergence in distribution)

16 Poisson approximation: the Stein-Chen method Theorem Let Z u {0, 1}, u V, X = u V Z u. Denote π u = E(Z u ), λ = E(X ) = u V π u. Assume {Z uv } u,v V,v u such that u V, P({Z uv } v u ) = P({Z v } v u Z u = 1). Then: d var (X, Poisson(λ)) 2 min(1, 1/λ) u V π u π u + E Z uv Z v v u

17 Applications Proposition (Binomial approximation) One has for all n, λ n: d var (Bin(n, λ/n), Poisson(λ)) 2 min(1, λ) λ n Proposition (Isolated nodes) In G(n, p) with np = log(n) + c, noting λ = n(1 p) n 1 e c and X : number of isolated nodes, then d var (X, Poisson(λ)) 2λ[1/n + p/(1 p)] = O(log(n)/n) Hence, lim n P(X = 0) = e e c

18 Stein-Chen method proof arguments Fact: for each λ > 0, A N, function f : N R defined by f (0) = 0, λf (j + 1) j f (j) = I A (j) Poi λ (A), j N is min(1, λ 1 ) Lipschitz Write P(X A) Poi λ (A) = E[λf (X + 1) Xf (X )] = π u E f (X + 1) f (1 + Z uv ) u V v u π u min(1, λ 1 )E Z v Z uv u V v V v u u V π u π u + E Z v Z uv v u

19 Connectivity final arguments Let A k = { connected component of size k}. By union bound, for p = Θ(log(n)/n), P(A 2 ) ( n 2) p(1 p) 2(n 2) O(p) = o(1) Similarly for k n/2, P(A k ) ( ) n k Tk p k 1 (1 p) k(n k) where T k : number of trees on [k] Cayley s theorem: T k = k k 2. Hence P(A k ) ( ) n k k k 2 p k 1 (1 p) k(n k) nk k! kk 2 p k 1 e pkn/2 Conclusion P( 2 k n/2 A k ) follows. 1 1 p k 2 k ek(1+log(np) np/2) 2 k n/2 P(A k ) 0 as n

20 Takeaway messages connectivity of Erdős-Rényi graphs informs behaviour of SIR epidemics on complete graph Emergence of giant component of size n(1 p ext ) as average degree crosses critical value 1 Full connectivity for average degree log(n) + O(1) Proof techniques: branching process approximation, Chernoff bounds; First and second moment methods; Poisson approximation via Stein-Chen method