The sharp threshold for jigsaw percolation in random graphs

Abstract

The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollobás, Riordan, Slivken and Smith.Crn(ln n) i−1 for some 2 ≤ i ≤ r or P 1 ≤ ln n Crn then whp G does not percolate;(ii) if P i ≥ Cr n(ln n) i−1 for every 2 ≤ i ≤ r and P 1 ≥ Cr ln n n , then whp G percolates.

The Size of the Giant Joint Component in a Binomial Random Double Graph

“…Several papers have been devoted to investigating when the process percolates, i.e., when every vertex is contained in the same part by the end of the process. So far the combination of various deterministic graphs with a binomial random graph [4,8] and the combination of two binomial random graphs [3,7] has been studied. In addition, extensions to multi-coloured random graphs [6] and random hypergraphs [2] exist.…”