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 function p = p(n) above which a giant component appears 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′. © 2009 Wiley Periodicals, Inc.
Random Structures and Algorithms
Bollobás, B., & Riordan, O. (2009). Clique percolation. Random Structures and Algorithms, 35 (3), 294-322. https://doi.org/10.1002/rsa.20270