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Erdös-Renyi Random Graphs The first random graph model was introduced by Erdös and Rényi in the late 1950's. To define the model, we begin with the set of vertices V = {1, 2,. .. n}. For 1 ≤ x < y ≤ n let η x,y be independent = 1 with probability p = λ/n and 0 otherwise. Let η y ,x = η x,y. If η x,y = 1 there is an edge from x to y. A large Erdös-Renyi random graph has a degree distribution that is Poisson with mean λ. However in many technological and social networks, the degree distribution p k follows a power law: p k ∼ Ck −α. Rick Durrett (Cornell) Random Graph Dynamics 2 / Figure: Sweden sex partners follow power law Rick Durrett (Cornell) Random Graph Dynamics 3 / 109 Fixed Degree Distributions Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model. Let d 1 ,. .. d n be independent and have P(d i = k) = p k. Since we want d i to be the degree of vertex i, we condition on E n = {d 1 + • • • + d n is even}. If the probability P(E 1) ∈ (0, 1) then P(E n) → 1/2 as n → ∞ so the conditioning will have little effect on the finite dimensional distributions. Rick Durrett (Cornell) Random Graph Dynamics 4 / Attach d i half-edges to vertex i and then pair the half-edges at random. This can produce parallel edges or self-loops, but if Ed 2 i < ∞ then with probability bounded away from 0 we get an ordinary graph. Rick Durrett (Cornell) Random Graph Dynamics 6 / Contact Process Consider the contact process on a power-law random graph. In this model infected individuals become healthy at rate 1 (and are again susceptible to the disease) susceptible individuals become infected at a rate λ times the number of infected neighbors. Pastor-Satorras and Vespigniani (2001a, 2001b, 2002) have made an extensive study of this model using mean-field methods (See Section 4.8.). Rick Durrett (Cornell) Random Graph Dynamics 7 / 109 Contact Process Conjectures Let λ c be the critical value for prolonged persistence. If λ > λ c there will be a quasi-stationary distribution with density ρ(λ) ∼ C (λ − λ c) β If α ≤ 3 then λ c = 0. If 3 < α < 4, λ c > 0 but the critical exponent β > 1. If α > 4 then λ c > 0 and β = 1. Problem. Berger, Borgs, Chayes, Saberi (2005) prove persistence for time exp(cn 1/2) for any λ > 0 when α = 3. Does it last for exp(cn)? Rick Durrett (Cornell) Random Graph Dynamics 8 / Voter models Vertex voter model. Each vertex x changes at rate 1. Pick a neighbor at random and set ξ(x) = ξ(y). Genealogical process jumps at rate 1. Stationary distribution π(x) = cd(x). Edge voter model. Each edge becomes active at rate 1. Flip a coin to give it an orientation (x, y) then set ξ t (x) = ξ t (y). Genealogical process of a site is a random walk that jumps to a randomly chosen neighbor at rate d(x). Stationary distribution is uniform. Rick Durrett (Cornell) Random Graph Dynamics 9 / 109 Rick Durrett (Cornell) Random Graph Dynamics 11 / 109 Rick Durrett (Cornell) Random Graph Dynamics 17 / 109 Rick Durre...

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In a homogeneously mixing population of E. coli, colicin-producing and colicin-sensitive strategies both may be evolutionarily stable for certain parameter ranges, with the outcome of competition determined by initial conditions. In contrast, in a spatially-structured population, there is a unique ESS for any given set of parameters; the outcome is determined by how effective allelopathy is in relation to its costs. Furthermore, in a spatially-structured environment, a dynamic equilibrium may be sustained among a colicin-sensitive type, a high colicin-producing type, and a "cheater" that expends less on colicin production but is resistant. Copyright 1997 Academic Press Limited