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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.