Listing All Maximal Cliques in Large Sparse Real-World Graphs

Eppstein, David; Löffler, Maarten; Strash, Darren

doi:10.1145/2543629

Cited by 112 publications

(54 citation statements)

References 41 publications

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“…The overall number of subsets to investigate is at most 2 d · n. Since d is often very small, this simple algorithm can serve as a basis for efficient CLIQUE algorithms. The approach based on degeneracy can also be used to enumerate all maximal cliques in O(3 d/3 · n) time [25].…”

Section: The Clique Problem From the Viewpoint Of Multivariate Algorimentioning

confidence: 99%

Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.

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Section: The Clique Problem From the Viewpoint Of Multivariate Algorimentioning

confidence: 99%

Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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“…Though these bounded time delay algorithms are theoretically efficient, Bron-Kerbosch-derived algorithms are much faster in practice. As noted by Conte et al [16], their algorithm (which is at present the fastest bounded time delay algorithm) is 3.7 times slower than the Bron-Kerbosch-derived algorithm by Eppstein et al [8] on sparse graphs, which is itself slower than the algorithm by Tomita et al [7] on dense and medium-density instances that we consider here. Even though the algorithm by Tomita et al [7] has worst-case exponential time, repeated experiments show that it is fast on a variety of benchmark instances [7,8,17,18,19].…”

Section: Introductionmentioning

confidence: 62%

Efficiently enumerating all maximal cliques with bit-parallelism

Segundo

Artieda

Strash

2018

Computers & Operations Research

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The maximal clique enumeration (MCE) problem has numerous applications in biology, chemistry, sociology, and graph modeling. Though this problem is well studied, most current research focuses on finding solutions in large sparse graphs or very dense graphs, while sacrificing efficiency on the most difficult medium-density benchmark instances that are representative of data sets often encountered in practice. We show that techniques that have been successfully applied to the maximum clique problem give significant speed gains over the state-of-the-art MCE algorithms on these instances. Specifically, we show that a simple greedy pivot selection based on a fixed maximum-degree first ordering of vertices, when combined with bit-parallelism, performs consistently better than the theoretical worst-case optimal pivoting of the state-of-the-art algorithms of Tomita et al. [Theoretical Computer Science, 2006] and Naudé [Theoretical Computer Science, 2016].Experiments show that our algorithm is faster than the worst-case optimal algorithm of Tomita et al. on 60 out of 74 standard structured and random benchmark instances: we solve 48 instances 1.2 to 2.2 times faster, and solve the remaining 12 instances 3.6 to 47.6 times faster. We also see consistent speed improvements over the algorithm of Naudé: solving 61 instances 1.2 to 2.4 times faster. To the best of our knowledge, we are the first to achieve such speed-ups compared to these state-of-the-art algorithms on these standard benchmarks.

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“…Empirically, we observe that the SCT is quite small. In the worst-case, | SCT (G)| = O(n3 α /3 ), which follows from arguments by Eppstein-Löeffler-Strash [12] and Tomita-Tanaka-Takahashi [29] (an exponential dependence is necessary because of the NP-hardness of maximum clique). We give a detailed description in §5…”

Section: Main Ideasmentioning

confidence: 92%

“…Thus, this method cannot scale to counting larger cliques, since the number of cliques is simply too large. Our main insight is that the method of pivoting, used to reduce recursion trees for maximal clique enumeration [7,12], can be applied to counting cliques of all sizes.…”

Section: Main Contributionsmentioning

confidence: 99%

The Power of Pivoting for Exact Clique Counting

Jain

Seshadhri

2020

Proceedings of the 13th International Conference on Web Search and Data Mining

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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.

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Listing All Maximal Cliques in Large Sparse Real-World Graphs

Cited by 112 publications

References 41 publications

Multivariate Algorithmics for Finding Cohesive Subnetworks

Multivariate Algorithmics for Finding Cohesive Subnetworks

Efficiently enumerating all maximal cliques with bit-parallelism

The Power of Pivoting for Exact Clique Counting

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