We study majority dynamics on the binomial random graph G(n, p) with p = ∕n and > n 1∕2 , for some large > 0. In this process, each vertex has a state in {−1, +1} and at each round every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. We show that with high probability the process reaches unanimity in at most four rounds. This confirms a conjecture of Benjamini et al.
Consider the triangle-free graph process, which starts from the empty graph on n vertices and a random ordering of the possible n 2 edges; the edges are added in this ordering provided the graph remains triangle free. We will show that there exists a constant c such that no copy of any fixed finite triangle-free graph on k vertices with at least ck edges asymptotically almost surely appears in the triangle-free graph process.
Abstract:Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2-balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs. C 2014 Wiley Periodicals, Inc. J. Graph Theory 79: [125][126][127][128][129][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144] 2015
A bootstrap percolation process on a graph with infection threshold r ≥ 1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever.Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Z d that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson, Luczak, Turova and Vallier (2012). In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model : each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.
Abstract. We investigate bootstrap percolation with infection threshold r > 1 on the binomial k-uniform random hypergraph H k (n, p) in the regime n −1 ≪ n k−2 p ≪ n −1/r , when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for k = 2, we show that the probability of failure decreases exponentially.
The stochastic Kronecker graph model introduced by Leskovec et al. is a random graph with vertex set Z n 2 , where two vertices u and v are connected with probabilityindependently of the presence or absence of any other edge, for fixed parameters 0 < α, β, γ < 1. They have shown empirically that the degree sequence resembles a power law degree distribution. In this paper we show that the stochastic Kronecker graph a.a.s. does not feature a power law degree distribution for any parameters 0 < α, β, γ < 1. In addition, we analyze the number of subgraphs present in the stochastic Kronecker graph and study the typical neighborhood of any given vertex.
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