Distributed discovery of large near-cliques

Given an undirected graph and 0≤ϵ≤1 , a set of nodes is called an ϵ -near clique if all but an ϵ 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 ϵ -near clique if there exists an ϵ^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 α∈ (0,1) . Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.