Opinion Forming in Erdos-Renyi Random Graph and Expanders
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Assume for a graph G=(V,E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the binomial random graph G_n,p and regular expanders. First we consider the behavior of the majority model in G_n,p with an initial random configuration, where each node is blue independently with probability p_b and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely n/n. Furthermore, we discuss the majority model is a `good' and `fast' density classifier on regular expanders. More precisely, we prove if the second-largest absolute eigenvalue of the adjacency matrix of an n-node Δ-regular graph is sufficiently smaller than Δ then the majority model by starting from (1/2-δ)n blue nodes (for an arbitrarily small constant δ>0) results in fully red configuration in sub-logarithmically many rounds. As a by-product of our results, we show Ramanujan graphs are asymptotically optimally immune, that is for an n-node Δ-regular Ramanujan graph if the initial number of blue nodes is s≤β n, the number of blue nodes in the next round is at most cs/Δ for some constants c,β>0. This settles an open problem by Peleg.
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