Current Flow Group Closeness Centrality for Complex Networks?

Li, Huan; Peng, Richard; Shan, Liren; Yi, Yuhao; Zhang, Zhongzhi

doi:10.1145/3308558.3313490

Cited by 30 publications

(18 citation statements)

References 55 publications

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“…It has been demonstrated in [26] that for individual nodes, the grounded centrality gives quite a different ranking for node importance, compared with many other node centrality measures, such as degree centrality, betweenness centrality, eigenvector centrality, closeness centrality, and information centrality. Thus, grounded node group centrality also deviates from previously proposed centrality measures for a node group, including betweenness centrality [32], [33], closeness centrality [34], [35], as well as current flow closeness centrality [36]. Therefore, it is of independent interest to study Problem 1, with an aim to find the k most important nodes as the set S * of grounded nodes corresponding to maximum λ(S * ).…”

Section: B Grounded Node Group Centralitymentioning

confidence: 99%

“…We thus use λ(S) to quantify the importance/centrality of the group of nodes in S, termed grounded node group centrality. There are numerous metrics for centrality of a group of nodes in a graph [31], based on structural or dynamical properties, including betweenness [32], [33], closeness centrality [34], [35], and current flow closeness centrality [36], and so on. However, since the criterion for importance of a set of nodes depends on specific applications [37], grounded node group centrality deviates from previous centrality metrics, even for an individual node [26].…”

Section: Related Workmentioning

confidence: 99%

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Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Wang¹,

Zhou²,

Li³

et al. 2021

Preprint

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For a connected graph G = (V, E) with n nodes, m edges, and Laplacian matrix L, a grounded Laplacian matrix L(S) of G is a (n − k) × (n − k) principal submatrix of L, obtained from L by deleting k rows and columns corresponding to k selected nodes forming a set S ⊆ V . The smallest eigenvalue λ(S) of L(S) plays a pivotal role in various dynamics defined on G. For example, λ(S) characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger λ(S) corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset S of fixed k ≪ n nodes, in order to maximize the smallest eigenvalue λ(S) of the grounded Laplacian matrix L(S). We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a naïve heuristic algorithm selecting one optimal node at each time for k iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naïve heuristic algorithm takes Õ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Õ(km), where Õ(•) notation suppresses the poly(log n) factors. We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness compared to baseline methods.

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Section: B Grounded Node Group Centralitymentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Wang¹,

Zhou²,

Li³

et al. 2021

Preprint

Self Cite

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show abstract

“…Moreover, for a nonnegative diagonal matrix X with at least one nonzero diagonal entry, we have the fact that every element of (L + X) −1 is positive [27,31,32].…”

Section: Graphs and Related Matricesmentioning

confidence: 99%

Maximizing Influence of Leaders in Social Networks

Zhou,

Zhang

2021

Preprint

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The operation of adding edges has been frequently used to the study of opinion dynamics in social networks for various purposes. In this paper, we consider the edge addition problem for the DeGroot model of opinion dynamics in a social network with nodes and edges, in the presence of a small number ≪ of competing leaders with binary opposing opinions 0 or 1. Concretely, we pose and investigate the problem of maximizing the equilibrium overall opinion by creating new edges in a candidate edge set, where each edge is incident to a 1-valued leader and a follower node. We show that the objective function is monotone and submodular. We then propose a simple greedy algorithm with an approximation factor (1 − 1 ) that approximately solves the problem in ( 3 ) time. Moreover, we provide a fast algorithm with a (1 − 1 − ) approximation ratio and ˜ ( −2 ) time complexity for any > 0, where ˜ (•) notation suppresses the poly(log ) factors. Extensive experiments demonstrate that our second approximate algorithm is efficient and effective, which scales to large networks with more than a million nodes.

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“…Chehreghani et al [13] provide an extensive analysis of several algorithms for group betweenness centrality estimation, and present a new algorithm based on an alternative definition of distance between a vertex and a group of vertices. Greedy approximation algorithms to maximize group centrality also exist for measures like closeness (by Bergamini et al [6]) and current-flow closeness (by Li et al [30]). Puzis et al [43] state an algorithm for group betweenness maximization that does not utilize submodular approxima- tion but relies on a branch-and-bound approach.…”

Section: Related Workmentioning

confidence: 99%

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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Current Flow Group Closeness Centrality for Complex Networks?

Cited by 30 publications

References 55 publications

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Maximizing Influence of Leaders in Social Networks

Group Centrality Maximization for Large-scale Graphs

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