Parallel local graph clustering

Abstract

Graph clustering has many important applications in computing, but due to growing sizes of graphs, even traditionally fast clustering methods such as spectral partitioning can be computationally expensive for real-world graphs of interest. Motivated partly by this, so-called local algorithms for graph clustering have received significant interest due to the fact that they can find good clusters in a graph with work proportional to the size of the cluster rather than that of the entire graph. This feature has proven to be crucial in making such graph clustering and many of its downstream applications efficient in practice. While local clustering algorithms are already faster than traditional algorithms that touch the entire graph, they are sequential and there is an opportunity to make them even more efficient via parallelization. In this paper, we show how to parallelize many of these algorithms in the shared-memory multicore setting, and we analyze the parallel complexity of these algorithms. We present comprehensive experiments on large-scale graphs showing that our parallel algorithms achieve good parallel speedups on a modern multicore machine, thus significantly speeding up the analysis of local graph clusters in the very large-scale setting.

Efficient Estimation of Heat Kernel PageRank for Local Clustering

“…Existing heat-kernel-based algorithms [10,12,16,21] all adopt a two-phase approach. In particular, they first compute an approximate HKPR vector ρ s for s, and then perform a sweep as follows:…”

“…Return the set S * i with the smallest conductance among the ones that have been inspected. It is shown in [21,42] that the above sweep can be conducted in O (|S * | · log |S * |) time, assuming that ρ s is given in a sparse representation with O(|S * |) entries. In contrast, the computation of ρ s is much more costly, and hence, has been the main subject of research in existing work [10,12,16,21].…”

“…Our solution for local clustering is based on heat kernel PageRank, and it follows the same two-phase framework in the existing work [7,8,10,12,16,21]. That is, we also compute an approximate HKPR vector ρ s for s, and then conduct a sweep on ρ s .…”

“…That is, we also compute an approximate HKPR vector ρ s for s, and then conduct a sweep on ρ s . However, we require that ρ s should be a (d, ϵ r , δ )-approximate HKPR vector, which is a criterion not considered in any existing work [7,8,10,12,16,21]. Definition 1.…”

“…In recent work [21], Shun et al study parallel implementations for Nibble, PR-Nibble, ClusterHKPR, and HK-Relax, respectively, and are able to achieve significant speedup on a machine with 40 cores. We believe that our algorithms may also exploit parallelism for higher efficiency, but a thorough treatment of this problem is beyond the scope of this paper.…”