@MISC{Bollobás08cliquepercolation, author = {Béla Bollobás and Oliver Riordan}, title = {Clique percolation}, year = {2008} }

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Abstract

Derényi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph G generated by some rule, form an auxiliary graph G ′ whose vertices are the k-cliques of G, in which two vertices are joined if the corresponding cliques share k − 1 vertices. They considered in particular the case where G = G(n, p), and found heuristically the threshold for a giant component to appear in G ′. Here we give a rigorous proof of this result, as well as many extensions. The model turns out to be very interesting due to the essential global dependence present in G ′. 1 Cliques sharing vertices Fix k ≥ 2 and 1 ≤ ℓ ≤ k − 1. Given a graph G, let Gk,ℓ be the graph whose vertex set is the set of all copies of Kk in G, in which two vertices are adjacent if the corresponding copies of Kk share at least ℓ vertices. Starting from a random graph G = G(n, p), our aim is to study percolation in the corresponding graph, i.e., to find for which values of p there is a ‘giant ’ component in Gk,ℓ p, containing a positive fraction of the vertices of Gk,ℓ p. For ℓ = k − 1, this question was proposed by Derényi, Palla and Vicsek [10], motivated by the study of ‘communities ’ in real-world networks, but independent of the motivation, we consider it to be an extremely natural question in the theory of random graphs. Indeed, it is perhaps the most natural example of dependent percolation arising out of the model G(n, p). As we shall see in a moment, it is not too hard to guess the answer; simple G k,ℓ p heuristic derivations based on the local analysis of G k,ℓ p were given in [10] and may well have is not well approximated by a graph with by Palla, Derényi and Vicsek [15]. Note, however, that G k,ℓ