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Random subgraphs of the 2D Hamming graph: the supercritical phase

By R van der Hofstad and Malwina Luczak

Abstract

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p=(1+ε)/2(n-1) for some ε∈ R. In [4,5], the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for ε≤ Λ V-1/3, where Λ > 0 is a constant and V=n2 denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ε>> (log V)1/3 V-1/3, then the largest connected component has size close to 2ε V with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor (log V)1/3, this identifies the size of the largest connected component all the way down to the critical p window

Topics: H Social Sciences (General)
Publisher: London school of economics and political science
Year: 2007
OAI identifier: oai:eprints.lse.ac.uk:6801
Provided by: LSE Research Online
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