Parallel Clique Counting and Peeling Algorithms
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Dense subgraphs capture strong communities in social networks and entities possessing strong interactions in biological networks. In particular, k-clique counting and listing have applications in identifying important actors in a graph. However, finding k-cliques is computationally expensive, and thus it is important to have fast parallel algorithms. We present a new parallel algorithm for k-clique counting that has polylogarithmic span and is work-efficient with respect to the well-known sequential algorithm for k-clique listing by Chiba and Nishizeki. Our algorithm can be extended to support listing and enumeration, and is based on computing low out-degree orientations. We present a new linear-work and polylogarithmic span algorithm for computing such orientations, and new parallel algorithms for producing unbiased estimations of clique counts. Finally, we design new parallel work-efficient algorithms for approximating the k-clique densest subgraph. Our first algorithm gives a 1/k-approximation and is based on iteratively peeling vertices with the lowest clique counts; our algorithm is work-efficient, but we prove that this process is P-complete and hence does not have polylogarithmic span. Our second algorithm gives a 1/(k(1+ϵ))-approximation, is work-efficient, and has polylogarithmic span. In addition, we implement these algorithms and propose optimizations. On a 60-core machine, we achieve 13.23-38.99x and 1.19-13.76x self-relative parallel speedup for k-clique counting and k-clique densest subgraph, respectively. Compared to the state-of-the-art parallel k-clique counting algorithms, we achieve a 1.31-9.88x speedup, and compared to existing implementations of k-clique densest subgraph, we achieve a 1.01-11.83x speedup. We are able to compute the 4-clique counts on the largest publicly-available graph with over two hundred billion edges.
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