Computing Top-<i>k</i> Closeness Centrality Faster in Unweighted Graphs

Bergamini, Elisabetta; Borassi, Michele; Crescenzi, Pierluigi; Marino, Andrea; Meyerhenke, Henning

doi:10.1137/1.9781611974317.6

Cited by 27 publications

(75 citation statements)

References 26 publications

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“…In this section, we evaluate the performance of our algorithms against the state-of-the-art greedy algorithm of Bergamini et al [12]. 4 As mentioned in Section I, it has been shown empirically that the solution quality yielded by the greedy algorithm is often nearly-optimal. We evaluate two variants, LS and LS-restrict (see Section II-D2), of our Local-Swap algorithm, and three variants, GS, GS-local (see Section II-D3) and GS-extended (see Section II-D4) of our Grow-Shrink algorithm.…”

Section: Methodsmentioning

confidence: 99%

Local Search for Group Closeness Maximization on Big Graphs

Angriman

Grinten

Meyerhenke

2019

2019 IEEE International Conference on Big Data (Big Data)

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In network analysis and graph mining, closeness centrality is a popular measure to infer the importance of a vertex. Computing closeness efficiently for individual vertices received considerable attention. The N P-hard problem of group closeness maximization, in turn, is more challenging: the objective is to find a vertex group that is central as a whole and state-ofthe-art heuristics for it do not scale to very big graphs yet.In this paper, we present new local search heuristics for group closeness maximization. By using randomized approximation techniques and dynamic data structures, our algorithms are often able to perform locally optimal decisions efficiently. The final result is a group with high (but not optimal) closeness centrality.We compare our algorithms to the current state-of-the-art greedy heuristic both on weighted and on unweighted real-world graphs. For graphs with hundreds of millions of edges, our local search algorithms take only around ten minutes, while greedy requires more than ten hours. Overall, our new algorithms are between one and two orders of magnitude faster, depending on the desired group size and solution quality. For example, on weighted graphs and k = 10, our algorithms yield solutions of 12.4% higher quality, while also being 793.6× faster. For unweighted graphs and k = 10, we achieve solutions within 99.4% of the state-of-the-art quality while being 127.8× faster.1 While this greedy algorithm was claimed to have a bounded approximation quality (e. g., in [7], [12]), the proof of this bound relied on the assumption that C is submodular. A recent update [13] to the conference version [12] revealed that, in fact, C is not submodular. We are not aware of any approximation algorithm for group closeness that scales to large graphs.

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Section: Methodsmentioning

confidence: 99%

Local Search for Group Closeness Maximization on Big Graphs

Angriman

Grinten

Meyerhenke

2019

2019 IEEE International Conference on Big Data (Big Data)

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“…4 Hence, we adapt the ideas of the top-k Katz ranking algorithm that was introduced in [49] to the case of GED(S, x). 5 Applied to the marginal gain of GED-Walk, the main ingredients of this algorithm are families L (S, x) and U (S, x) of lower and upper bounds on GED(S, x), satisfying the following definition:…”

Section: Maximizingmentioning

confidence: 99%

“…Top-1 algorithms have been developed for multiple centrality measures, e. g., in [5]. 5 Note that the vertex that maximizes GED(S, x) = GED(S ∪ {x}) − GED(S) is exactly the vertex that maximizes GED(S ∪ {x}). However, algorithmically, it is advantageous to deal with GED(S ∪{x}), as it allows us to construct a lazy greedy algorithm (see Section 3.2.2).…”

Section: Maximizingmentioning

confidence: 99%

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Group Centrality Maximization for Large-scale Graphs

Angriman¹,

Grinten²,

Bojchevski³

et al. 2020

2020 Proceedings of the Twenty-Second Workshop on Algorithm Engineering and Experiments (ALENEX)

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The study of vertex centrality measures is a key aspect of network analysis. Naturally, such centrality measures have been generalized to groups of vertices; for popular measures it was shown that the problem of finding the most central group is N P-hard. As a result, approximation algorithms to maximize group centralities were introduced recently. Despite a nearly-linear running time, approximation algorithms for group betweenness and (to a lesser extent) group closeness are rather slow on large networks due to high constant overheads.That is why we introduce GED-Walk centrality, a new submodular group centrality measure inspired by Katz centrality. In contrast to closeness and betweenness, it considers walks of any length rather than shortest paths, with shorter walks having a higher contribution. We define algorithms that (i) efficiently approximate the GED-Walk score of a given group and (ii) efficiently approximate the (proved to be N P-hard) problem of finding a group with highest GED-Walk score.Experiments on several real-world datasets show that scores obtained by GED-Walk improve performance on common graph mining tasks such as collective classification and graph-level classification. An evaluation of empirical running times demonstrates that maximizing GED-Walk (in approximation) is two orders of magnitude faster compared to group betweenness approximation and for group sizes ≤ 100 one to two orders faster than group closeness approximation. For graphs with tens of millions of edges, approximate GED-Walk maximization typically needs less than one minute. Furthermore, our experiments suggest that the maximization algorithms scale linearly with the size of the input graph and the size of the group.2 Katz centrality is sometimes alternatively defined based on walks ending at a given vertex; from an algorithmic perspective, however, this difference is irrelevant.

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“…Centrality theory [8][9][10][11][12], diffusion models [13], heat diffusion theory [14], evidence theory [15] etc., are the most frequently used techniques for obtaining top-K influential nodes in a network.…”

Section: Related Workmentioning

confidence: 99%