From Stars to Diamonds: Counting and Listing Almost Complete Subgraphs in Large Networks

Listing dense subgraphs is a fundamental task with a variety of network analytics applications. A lot of research has been done focusing on $k$-cliques, i.e. complete subgraphs on $k$ nodes. However, requiring complete connectivity between the nodes of a subgraph may be too restrictive in many real applications. Hence, in this paper, we consider a natural relaxation of cliques, called $k$-diamonds and defined as cliques of size $k$ with one missing edge. We first provide a sequential algorithm that, in $O(nm<^>{(k-1)/2})$ time, counts and lists all the $k$-diamonds in large graphs, for any constant $k \geq 4$. A parallel extension of the sequential algorithm is then proposed and analyzed in a MapReduce-style model, achieving the same local and total space usage of the state-of-the-art algorithms for $k$-cliques. The running time is optimal on dense graphs and $O(\sqrt{m})$ larger than $k$-clique counting if the graph is sparse. Our algorithms compute induced diamonds by analyzing the structure of directed stars formed by the graph nodes and their neighbors.