Clique percolation

TL;DR

This paper rigorously proves the threshold for giant component emergence in a dependent percolation model based on clique sharing in random graphs, extending previous heuristic results and exploring the model's global dependence.

Contribution

It provides a rigorous proof of the clique percolation threshold in random graphs and extends the analysis to various model extensions.

Findings

Established the threshold for giant component formation in clique percolation.

Demonstrated the global dependence effects in the auxiliary graph.

Extended the model to include multiple variants and conditions.

Abstract

Derenyi, 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 $G'$.