Cliquecounting is a fundamental problem that has application in many areas eg. dense subgraph discovery, community detection, spam detection, etc. The problem of kcliquecounting is difficult because as k increases, the number of kcliques goes up exponentially. Enumeration algorithms (even parallel ones) fail to count kcliques 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 stateoftheart and was able to give exact counts of all kcliques in a large number of graphs. However, the clique counts of some graphs (for example, comlj) 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 realworld graphs to achieve faster cliquecounting. 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 kclique density in a subgraph in terms of its edge density. However, this stopping condition is based on a worstcase graph that does not reflect the nature of realworld graphs. Using techniques formore »
Mining Largest Maximal QuasiCliques
Quasicliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasicliques from a graph is a robust way to detect densely connected structures with applications in bioinformatics and social network analysis. However, enumerating quasicliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top k degreebased quasicliques and make the following contributions: (1) we show that even the problem of detecting whether a given quasiclique is maximal (i.e., not contained within another quasiclique) is NPhard. (2) We present a novel heuristic algorithm K ernel QC to enumerate the k largest quasicliques in a graph. Our method is based on identifying kernels of extremely dense subgraphs within a graph, followed by growing subgraphs around these kernels, to arrive at quasicliques with the required densities. (3) Experimental results show that our algorithm accurately enumerates quasicliques from a graph, is much faster than current stateoftheart methods for quasiclique enumeration (often more than three orders of magnitude faster), and can scale to larger graphs than current methods.
 Award ID(s):
 1725702
 Publication Date:
 NSFPAR ID:
 10309138
 Journal Name:
 ACM Transactions on Knowledge Discovery from Data
 Volume:
 15
 Issue:
 5
 ISSN:
 15564681
 Sponsoring Org:
 National Science Foundation
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