The Power of Pivoting for Exact Clique Counting

Abstract

Clique counting is a fundamental task in network analysis, and even the simplest setting of 3-cliques (triangles) has been the center of much recent research. Getting the count of k-cliques for larger k is algorithmically challenging, due to the exponential blowup in the search space of large cliques. But a number of recent applications (especially for community detection or clustering) use larger clique counts. Moreover, one often desires local counts, the number of k-cliques per vertex/edge.Our main result is Pivoter, an algorithm that exactly counts the number of k-cliques, for all values of k. It is surprisingly effective in practice, and is able to get clique counts of graphs that were beyond the reach of previous work. For example, Pivoter gets all clique counts in a social network with a 100M edges within two hours on a commodity machine. Previous parallel algorithms do not terminate in days. Pivoter can also feasibly get local per-vertex and per-edge k-clique counts (for all k) for many public data sets with tens of millions of edges. To the best of our knowledge, this is the first algorithm that achieves such results.The main insight is the construction of a Succinct Clique Tree (SCT) that stores a compressed unique representation of all cliques in an input graph. It is built using a technique called pivoting, a classic approach by Bron-Kerbosch to reduce the recursion tree of backtracking algorithms for maximal cliques. Remarkably, the SCT can be built without actually enumerating all cliques, and provides a succinct data structure from which exact clique statistics (k-clique counts, local counts) can be read off efficiently.

Parallel Clique Counting and Peeling Algorithms

“…We additionally compare our counting algorithms to Jain and Seshadhri's PIVOTER algorithm [28], Mhedhbi and Salihoglu's worst-case optimal join algorithm (WCO) [35], Lai et al's implementation of a binary join algorithm (BINARYJOIN) [30], and Pinar et al's ESCAPE algorithm [41]. Note that PIVOTER is designed for counting all cliques, and the latter three algorithms are designed for general subgraph counting.…”

“…We also parallelize an external-memory algorithm by Goodrich and Pszona [25] and obtain the same complexity bounds. We believe that our parallel ranking algorithms may be of independent interest, as many other subgraph finding algorithms use low out-degree orderings (e.g., [25,41,28]).…”

“…We show that on a variety of realworld graphs and different k, our k-clique counting algorithm achieves 1.31-9.88x speedup over the state-of-the-art parallel KCLIST algorithm [15] and self-relative speedups of 13.23-38.99x. We also compared our k-clique counting algorithm to other parallel k-clique counting implementations including Jain and Seshadhri's PIVOTER [28], Mhedhbi and Salihoglu's worst-case optimal join algorithm (WCO) [35], Lai et al's implementation of a binary join algorithm (BINARYJOIN) [30], and Pinar et al's ESCAPE [41], and demonstrate speedups of up to several orders of magnitude.…”

Sandslash

“…Hand-optimized GPM applications [4,17,30,47] use algorithmic insight to prune the search tree. Table 2a (right) lists optimizations applicable to each application, and Table 2b (right) lists those that are supported by GPM systems.…”

“…This is useful when both patterns are being searched for or when one pattern is more efficient to search for than the other. This typically requires a local count [30] (micro-level count [4]) of embeddings associated with a single vertex or edge instead of a global count (macro-level count) of embeddings that match the pattern.…”