Finding Dense Subgraphs with Size Bounds

Andersen, Reid; Chellapilla, Kumar

doi:10.1007/978-3-540-95995-3_3

Cited by 173 publications

(207 citation statements)

References 15 publications

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“…Since dense subgraph discovery constitutes a main research topic in graph analysis, a wide variety of related methods exists: heuristics [49,52,54], algorithmic contributions on NP-hard formulations [5,12,22,40, ?] and poly-time solvable formulations [18, 40, ?].…”

Section: Related Workmentioning

confidence: 99%

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The K-clique Densest Subgraph Problem

Tsourakakis

2015

Proceedings of the 24th International Conference on World Wide Web

215

160

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Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining" [10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks.In this work, we introduce the k-clique densest subgraph problem, k ≥ 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k = 2. For k = 3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem: given a graph G(V, E), find a subset of vertices S * such that, where t(S) is the number of triangles induced by the set S.On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the kclique densest subgraph problem. Furthermore, we propose an efficient 1 k -approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8,18] for k = 2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k ≥ 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k = 2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k = 2 and k = 3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 kguarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-cliques. Finally, we evaluate our proposed method on a popular graph mining application.

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Section: Related Workmentioning

confidence: 99%

“…Khuller and Saha [40] provide a linear time -approximation algorithm for the case of directed graphs among other contributions. Two interesting variations of the DkS problem were introduced by Andersen and Chellapilla [5]. The two problems ask for the set S that maximizes the density subject to |S| ≤ k (DamkS) and |S| ≥ k (DalkS).…”

Section: Related Workmentioning

confidence: 99%

The K-clique Densest Subgraph Problem

Tsourakakis

2015

Proceedings of the 24th International Conference on World Wide Web

215

160

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“…Many graph problems like maximal clique finding [4], dense subgraph discovery [2], and betweenness approximation [18] use k-core decomposition as a subroutine.…”

Section: Related Workmentioning

confidence: 99%

“…Finding k-cores in a graph is a fundamental operation for many graph algorithms. k-core is commonly used as part of community detection algorithms [16], as well as for finding dense components in graphs [2,4,19], as a filtering step for finding large cliques (as a k-clique is also a k-1-core), and for large-scale network visualization [1].…”

Section: Introductionmentioning

confidence: 99%

Streaming algorithms for k-core decomposition

et al. 2013

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A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-Hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for streaming graph data. In this paper, we propose the first incremental k-core decomposition algorithms for streaming graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have to be updated, and efficiently process this subgraph to update the k-core decomposition. Our results show a significant reduction in run-time compared to non-incremental alternatives. We show the efficiency of our algorithms on different types of real and synthetic graphs, at different scales. For a graph of 16 million vertices, we observe speedups reaching a million times, relative to the non-incremental algorithms.

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“…Finding clusters is a data-intensive operation and it can easily become compute-intensive as well, depending on the heuristics used. The problem is equivalent to the problem of maximal, variable-sized dense subgraphs (or quasi-cliques), and theoretically speaking, several of the corresponding optimization problems are computationally hard problems [1,11,16] or with large degree polynomial methods [25,26]. Therefore, faster approximation heuristics need to be used in practice.…”

Section: Introductionmentioning

confidence: 99%

An OpenMP algorithm and implementation for clustering biological graphs

Chapman

Kalyanaraman

2011

Proceedings of the 1st Workshop on Irregular Applications: Architectures and Algorithms

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Graph algorithms on parallel architectures present an interesting case study for irregular applications. Among the graph algorithms popular in scientific computing, graph clustering or community detection has numerous applications in computational biology. However, this operation also poses serious computational challenges because of irregular memory access patterns, large memory requirements, and their dependence on other auxiliary (also irregular) data structures to supplement processing. In this paper, we address the problem of graph clustering on shared memory machines. We present a new OpenMP-based parallel algorithm called pClust-sm, which uses adjacency lists, hash tables and unionfind data structures in parallel. The algorithm improves both the asymptotic runtime and memory complexities of a previous serial implementation. Preliminary results show that this algorithm can scale up to 8 threads (cores) of a shared memory machine on a real world metagenomics input graph with 1.2M vertices and 100M edges. More importantly, the new implementation drastically reduces the time to solution from the order of several hours to just over 4 minutes, and in addition, it enhances the problem size reach by at least one order of magnitude.

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Finding Dense Subgraphs with Size Bounds

Cited by 173 publications

References 15 publications

The K-clique Densest Subgraph Problem

The K-clique Densest Subgraph Problem

Streaming algorithms for k-core decomposition

An OpenMP algorithm and implementation for clustering biological graphs

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