Clique-counting is a fundamental problem that has application in many areas eg. dense subgraph discovery, community detection, spam detection, etc. The problem of k-clique-counting is difficult because as k increases, the number of k-cliques goes up exponentially. Enumeration algorithms (even parallel ones) fail to count k-cliques beyond a small k. Approximation algorithms, like TuránShadow have been shown to perform well upto k = 10, but are inefficient for larger cliques. The recently proposed Pivoter algorithm significantly improved the state-of-the-art and was able to give exact counts of all k-cliques in a large number of graphs. However, the clique counts of some graphs (for example, com-lj) are still out of reach of these algorithms.
We revisit the TuránShadow algorithm and propose a generalized framework called YACC that leverages several insights about real-world graphs to achieve faster clique-counting. The bottleneck in TuránShadow is a recursive subroutine whose stopping condition is based on a classic result from extremal combinatorics called Turán's theorem. This theorem gives a lower bound for the k-clique density in a subgraph in terms of its edge density. However, this stopping condition is based on a worst-case graph that does not reflect the nature of real-world graphs. Using techniques for quickly discovering dense subgraphs, we relax the stopping condition in a systematic way such that we get a smaller recursion tree while still maintaining the guarantees provided by TuránShadow. We deploy our algorithm on several real-world data sets and show that YACC reduces the size of the recursion tree and the running time by over an order of magnitude. Using YACC, we are able to obtain clique counts for several graphs for which clique-counting was infeasible before, including com-lj.
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Parallel Clique-Like Subgraph Counting and Listing
Cliques and clique-like subgraphs (e.g., quasi-cliques) are important dense structures whose counting or listing are essential in applications like complex network analysis and community detection. These problems are usually solved by divide and conquer, where a task over a big graph can be recursively divided into subtasks over smaller subgraphs whose search spaces are disjoint. This divisible algorithmic paradigm brings enormous potential for parallelism, since different subtasks can run concurrently to drastically reduce the overall running time.
In this paper, we explore this potential by proposing a unified framework for counting and listing clique-like subgraphs. We study how to divide and distribute the counting and listing tasks, and meanwhile, to balance the assigned workloads of each thread dynamically. Four applications are studied under our parallel framework, i.e., triangle counting, clique counting, maximal clique listing and quasi-clique listing. Extensive experiments are conducted which demonstrate that our solution achieves an ideal speedup on various real graph datasets.
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- Award ID(s):
- 1755464
- NSF-PAR ID:
- 10140006
- Date Published:
- Journal Name:
- Proceedings of the 38th International Conference on Conceptual Modeling (ER)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-33223-5_40
