Scalable Betweenness Centrality Maximization via Sampling

Mahmoody, Ahmad; Tsourakakis, Charalampos E.; Upfal, Eli

doi:10.1145/2939672.2939869

Cited by 39 publications

(41 citation statements)

References 57 publications

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“…The problem of finding a central group of nodes has also been considered for betweenness centrality, for which sampling-based approximation algorithms have been proposed [19,28].…”

Section: Related Workmentioning

confidence: 99%

Scaling up Group Closeness Maximization

Bergamini¹,

Gonser²,

Meyerhenke³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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Closeness is a widely-used centrality measure in social network analysis. For a node it indicates the inverse average shortest-path distance to the other nodes of the network. While the identification of the k nodes with highest closeness received significant attention, many applications are actually interested in finding a group of nodes that is central as a whole. For this problem, only recently a greedy algorithm with approximation ratio (1 − 1/e) has been proposed [Chen et al., ADC 2016]. Since this algorithm's running time is still expensive for large networks, a heuristic without approximation guarantee has also been proposed in the same paper.In the present paper we develop new techniques to speed up the greedy algorithm without losing its theoretical guarantee. Compared to a straightforward implementation, our approach is orders of magnitude faster and, compared to the heuristic proposed by Chen et al., we always find a solution with better quality in a comparable running time in our experiments.Our method Greedy++ allows us to approximate the group with maximum closeness on networks with up to hundreds of millions of edges in minutes or at most a few hours. To have the same theoretical guarantee, the greedy approach by [Chen et al., ADC 2016] would take several days already on networks with hundreds of thousands of edges. In a comparison with the optimum, our experiments show that the solution found by Greedy++ is actually much better than the theoretical guarantee. Over all tested networks, the empirical approximation ratio is never lower than 0.97.Finally, we study for the first time the correlation between the top-k nodes with highest individual closeness and an approximation of the most central group in large complex networks. Our results show that the overlap between the two is relatively small, which indicates empirically the need to distinguish clearly between the two problems.

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“…The problem of finding a central group of nodes has also been considered for betweenness centrality, for which sampling-based approximation algorithms have been proposed [19,28].…”

Section: Related Workmentioning

confidence: 99%

Scaling up Group Closeness Maximization

Bergamini¹,

Gonser²,

Meyerhenke³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…To obtain a (1 − 1 e − )-approximation, their algorithms need at least Ω(mn −2 ) running time according to Theorem 2 of [MTU16]. Given the assumption that the maximum betweenness centrality among all sets of k vertices is Θ(n 2 ), the algorithm in [MTU16] is able to obtain a solution with the same approximation ratio in O((m + n)k −2 log n) time.…”

Section: Related Workmentioning

confidence: 99%

Maximizing the Number of Spanning Trees in a Connected Graph

Patterson

et al. 2020

IEEE Trans. Inform. Theory

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We study the problem of maximizing the number of spanning trees in a connected graph by adding at most k edges from a given candidate edge set, a problem that has applications in domains including robotics, network science, and cooperative control. By Kirchhoff's matrix-tree theorem, this problem is equivalent to maximizing the determinant of an SDDM matrix. We give both algorithmic and hardness results for this problem: * Stacy Patterson is supported in part by NSF grants CNS-1553340 and CNS-1527287.We study the problem of maximizing the number of spanning trees in a weighted connected graph G by adding at most k edges from a given candidate edge set. By Kirchhoff's matrixtree theorem [Kir47], the number of spanning trees in G is equivalent to the determinant of a minor of the graph Laplacian L. Thus, an equivalent problem is to maximize the determinant of a minor of L, or, more generally, to maximize the determinant of an SDDM matrix. The problem of maximizing the number of spanning trees, and the related problem of maximizing the determinant of an SDDM matrix, have applications in a wide variety of problem domains. We briefly review some of these applications below.In robotics, the problem of maximizing the number of spanning trees has been applied in graph-based Simultaneous Localization and Mapping (SLAM). In graph-based SLAM [TM06], each vertex corresponds to a robot's pose or position, and edges correspond to relative measurements between poses. The graph is used to estimate the most likely pose configurations. Since measurements can be noisy, a larger number of measurements results in a more accurate estimate. The problem of selecting which k measurements to add to a SLAM pose graph to most improve the estimate has been recast as a problem of selecting the k edges to add to the graph that maximize the number of spanning trees [KHD15, KHD16, KSHD16a, KSHD16b]. We note that the complexity of the estimation problem increases with the number of measurements, and so sparse, well-connected pose graphs are desirable [DK06]. Thus, one expects k to be moderately sized with respect to the number of vertices.In network science, the number of spanning trees has been studied as a measure of reliability in communication networks, where reliability is defined as the probability that every pair of vertices can communicate [Myr96]. Thus, network reliability can be improved by adding edges that most increase the number of spanning trees [FL01]. The number of spanning trees has also been used as a predictor of the spread of information in social networks [BAE11], with a larger number of spanning trees corresponding to better information propagation.In the field of cooperative control, the log-number of spanning trees has been shown to capture the robustness of linear consensus algorithms. Specifically, the log-number of spanning trees quantifies the network entropy, a measure of how well the agents in the network maintain agreement when subject to external stochastic disturbances [SM14, dBCM15, ZEP11]. Thus, the problem of ...

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“…There exist various measures for centrality of a group of vertices, based on graph structure or dynamic processes, such as betweenness [DEPZ09,FS11,Yos14,MTU16], absorbing random-walk centrality [LYHC14, MMG15, ZLX + 17], and grounding centrality [PS14,CHBP17]. Since the criterion for importance of a vertex group is application dependent [GTLY14], many previous works focus on selecting (or deleting) a group of k vertices (for some given k) in order to optimize related quantities.…”

Section: Related Workmentioning

confidence: 99%

Current Flow Group Closeness Centrality for Complex Networks?

Peng

Shan

et al. 2019

The World Wide Web Conference

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Current flow closeness centrality (CFCC) has a better discriminating ability than the ordinary closeness centrality based on shortest paths. In this paper, we extend this notion to a group of vertices in a weighted graph, and then study the problem of finding a subset S of k vertices to maximize its CFCC C(S), both theoretically and experimentally. We show that the problem is NP-hard, but propose two greedy algorithms for minimizing the reciprocal of C(S) with provable guarantees using the monotoncity and supermodularity. The first is a deterministic algorithm with an approximation factor (1 − k k−1 · 1 e ) and cubic running time; while the second is a randomized algorithm with a (1 − k k−1 · 1 e − ǫ)-approximation and nearly-linear running time for any ǫ > 0. Extensive experiments on model and real networks demonstrate that our algorithms are effective and efficient, with the second algorithm being scalable to massive networks with more than a million vertices.

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Scalable Betweenness Centrality Maximization via Sampling

Cited by 39 publications

References 57 publications

Scaling up Group Closeness Maximization

Scaling up Group Closeness Maximization

Maximizing the Number of Spanning Trees in a Connected Graph

Current Flow Group Closeness Centrality for Complex Networks?

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