Distributed Discovery of Large Near-Cliques

Abstract

Given an undirected graph and $0\le\epsilon\le1$, a set of nodes is called $\epsilon$-near clique if all but an $\epsilon$ fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size $\epsilon$-near clique if there exists an $\epsilon^3$-near clique of linear size in the graph. The algorithm uses messages of $O(\log n)$ bits. The failure probability can be reduced to $n^{-\Omega(1)}$ in $O(\log n)$ time, and the algorithm also works if the graph contains a clique of size $\Omega(n/\log^{\alpha}\log n)$ for some $\alpha \in (0,1)$.