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Quasi-cliques are a type of dense subgraphs that generalize the notion of cliques, important for applications such as community/module detection in various social and biological networks. However, the existing quasi-clique definition and algorithms are only applicable to undirected graphs. In this paper, we generalize the concept of quasi-cliques to directed graphs by proposing $$(\gamma_1, \gamma_2)$$-quasi-cliques which have density requirements in both inbound and outbound directions of each vertex in a quasi-clique subgraph. An efficient recursive algorithm is proposed to find maximal $$(\gamma_1, \gamma_2)$$-quasi-cliques which integrates many effective pruning rules that are validated by ablation studies. We also study the finding of top-$$k$$ large quasi-cliques directly by bootstrapping the search from more compact quasi-cliques, to scale the mining to larger networks. The algorithms are parallelized with effective load balancing, and we demonstrate that they can scale up effectively with the number of CPU cores.
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YACC: A Framework Generalizing TuránShadow for Counting Large Cliques
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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- PAR ID:
- 10380841
- Date Published:
- Journal Name:
- SDM2022
- Page Range / eLocation ID:
- 684 - 692
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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