The Contact Process on Evolving Scale-free Networks Motivation and results Amitai Linker (Universidad de Chile) Joint work with Peter M¨ orters (University of Bath) Emmanuel Jacob (ENS Lyon)
The contact process
The contact process Given a finite or countably infinite connected graph G = ( V , E ), the contact ag Markov process { X t } t ≥ 0 with state space { 0 , 1 } V , where process is a c` adl` 0 − → healthy 1 − → infected 1
The contact process Given a finite or countably infinite connected graph G = ( V , E ), the contact ag Markov process { X t } t ≥ 0 with state space { 0 , 1 } V , where process is a c` adl` 0 − → healthy 1 − → infected 1
The contact process Given a finite or countably infinite connected graph G = ( V , E ), the contact ag Markov process { X t } t ≥ 0 with state space { 0 , 1 } V , where process is a c` adl` 0 − → healthy 1 − → infected If a vertex y is infected, it recovers at rate 1, while if it is healthy, it gets infected at rate � λ X t ( z ) z ∼ y where λ > 0 is called the infection rate. 1
The contact process Given a finite or countably infinite connected graph G = ( V , E ), the contact ag Markov process { X t } t ≥ 0 with state space { 0 , 1 } V , where process is a c` adl` 0 − → healthy 1 − → infected If a vertex y is infected, it recovers at rate 1, while if it is healthy, it gets infected at rate � λ X t ( z ) z ∼ y where λ > 0 is called the infection rate. The configuration { 0 } V is the only absorbing state, and the event of reaching is called extinction. 1
Phase transition 1 the distribution of X t given X 0 = { 1 } V , it can easily be seen that P t Calling P t 1 converges weakly to the upper invariant measure ¯ ν satisfying δ 0 ≤ µ ≤ ¯ ν for all invariant measure µ . 2
Phase transition 1 the distribution of X t given X 0 = { 1 } V , it can easily be seen that P t Calling P t 1 converges weakly to the upper invariant measure ¯ ν satisfying δ 0 ≤ µ ≤ ¯ ν for all invariant measure µ . One of the main features of the contact process on infinite graphs G is the existence of some 0 ≤ λ 0 ( G ) < ∞ such that • λ < λ 0 = ⇒ δ 0 = ¯ ν • λ > λ 0 = ⇒ δ 0 ⊥ ¯ ν 2
Phase transition 1 the distribution of X t given X 0 = { 1 } V , it can easily be seen that P t Calling P t 1 converges weakly to the upper invariant measure ¯ ν satisfying δ 0 ≤ µ ≤ ¯ ν for all invariant measure µ . One of the main features of the contact process on infinite graphs G is the existence of some 0 ≤ λ 0 ( G ) < ∞ such that • λ < λ 0 = ⇒ δ 0 = ¯ ν • λ > λ 0 = ⇒ δ 0 ⊥ ¯ ν Which is equivalent to λ > λ 0 ⇐ ⇒ for all x ∈ V , P x ( T ext = ∞ ) > 0 2
Phase transition Our starting point is the following theorem, by Pemantle: Theorem: Pemantle (1992) Let G be supercritical Galton-Watson tree with descendant distribution having heavy tails , then λ 0 ( G ) = 0. 3
Phase transition Our starting point is the following theorem, by Pemantle: Theorem: Pemantle (1992) Let G be supercritical Galton-Watson tree with descendant distribution having heavy tails , then λ 0 ( G ) = 0. 3
Phase transition The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? 4
Phase transition The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then � T ext > e Ck λ 2 � P x = 1 − o (1 / k ) . 4
Phase transition The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then � T ext > e Ck λ 2 � P x = 1 − o (1 / k ) . 4
Phase transition The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then � T ext > e Ck λ 2 � P x = 1 − o (1 / k ) . In this case, the central vertex is “powerful” if k λ 2 >> 1. In particular we notice that this definition depends on λ . 4
Finite networks
• For finite networks, { 0 , 1 } V is finite, so the process gets extinct a.s. in finite time. • If G is finite but sufficiently large, it seems “infinite” for a single infected vertex. 5
• For finite networks, { 0 , 1 } V is finite, so the process gets extinct a.s. in finite time. • If G is finite but sufficiently large, it seems “infinite” for a single infected vertex. • If { G n } n ∈ N is a sequence of finite graphs converging locally to some infinite graph G , then for n large we expect X t to behave differently on G n depending on whether λ < λ 0 ( G ) − ε or λ 0 ( G ) + ε < λ 5
New phase transition Even though G n is finite and P t 1 → δ 0 weakly, if λ > λ 0 the existence of ¯ ν for G is reflected into the existence of some ¯ ν n ⊥ δ 0 which acts as a virtual equilibrium for a long time. 6
New phase transition Even though G n is finite and P t 1 → δ 0 weakly, if λ > λ 0 the existence of ¯ ν for G is reflected into the existence of some ¯ ν n ⊥ δ 0 which acts as a virtual equilibrium for a long time. 1. We have Fast extinction if E 1 ( T ext ) → ∞ at most polynomially with | V n | . 2. We have Slow extinction if E 1 ( T ext ) → ∞ exponentially with | V n | . 6
New phase transition Even though G n is finite and P t 1 → δ 0 weakly, if λ > λ 0 the existence of ¯ ν for G is reflected into the existence of some ¯ ν n ⊥ δ 0 which acts as a virtual equilibrium for a long time. 1. We have Fast extinction if E 1 ( T ext ) → ∞ at most polynomially with | V n | . 2. We have Slow extinction if E 1 ( T ext ) → ∞ exponentially with | V n | . With slow extinction we can have metastability, where the density of infected individuals, ρ ( X t ) stabilizes around a fixed value (called the metastable density ) until an excursion takes the process to extinction. 6
Our main motivation Consider the configuration model G = ( V , E ) whose degree distribution follows a power law of parameter τ , that is, for any vertex x , � � ≍ k − τ deg ( x ) = k P 7
Our main motivation Consider the configuration model G = ( V , E ) whose degree distribution follows a power law of parameter τ , that is, for any vertex x , � � ≍ k − τ deg ( x ) = k P In 2002, Pastor-Satorras and Vespignani used mean-field approximations to predict: • If τ < 3, there is Slow extinction for all λ > 0 • If τ > 3, there is Fast extinction for λ small. 7
Our main motivation Consider the configuration model G = ( V , E ) whose degree distribution follows a power law of parameter τ , that is, for any vertex x , � � ≍ k − τ deg ( x ) = k P In 2002, Pastor-Satorras and Vespignani used mean-field approximations to predict: • If τ < 3, there is Slow extinction for all λ > 0 • If τ > 3, there is Fast extinction for λ small. What is more, if 2 < τ < 3 their method predicts a metastable density of the 1 3 − τ for λ small. form λ 7
The main motivation The prediction was wrong! Chaterjee and Durret (2009) proved that there is slow extinction for all values of τ and λ . Later, Mountford et al. (2013) proved that in this case the metastable density has the form λ e + o (1) where � 1 if 2 < τ < 2 . 5 3 − τ e = 2 τ − 3 if 2 . 5 < τ 8
• This correction shows that the instant loss of “connection memory” of the mean-field approximations can hurt the infection, destroying the effect studied by Pemantle. 9
• This correction shows that the instant loss of “connection memory” of the mean-field approximations can hurt the infection, destroying the effect studied by Pemantle. • On the other hand, if τ is small, there is some survival mechanism for the process which does not rely on local survival. 9
• This correction shows that the instant loss of “connection memory” of the mean-field approximations can hurt the infection, destroying the effect studied by Pemantle. • On the other hand, if τ is small, there is some survival mechanism for the process which does not rely on local survival. • We want to understand mild versions of this effect by running the C.P. on graphs that change over time, giving the “connection memory” some finite lifespan. 9
Our evolving scale-free network model
Our construction: • The set of vertices is { 1 , 2 , . . . , N } • Each pair of vertices i and j is initially connected independently with probability � i � 1 N , j N p N for some continuous, symmetric, decreasing function p : (0 , 1] 2 → R + such that � 1 p ( a , s ) ds ≍ a − γ 0 for some γ ∈ (0 , 1). 10
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