Distributed discovery of large near-cliques

Abstract

Given an undirected graph and $${0\le\epsilon\le1}$$ , a set of nodes is called an $${\epsilon}$$ -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 −Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) α ) for some $${\alpha \in (0,1)}$$ . Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.