Current Flow Group Closeness Centrality for Complex Networks
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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 the notion of CFCC to a group of vertices in a weighted graph. For a graph with n vertices and m edges, the CFCC C(S) for a vertex group S is equal to the ratio of n to the sum of effective resistances from S to all other vertices. We then study the problem of finding a group S^* of k vertices, so that the CFCC C(S^*) is maximized. We alternatively solve this problem by minimizing the reciprocal of C(S^*). We show that the problem is NP-hard, and prove that the objective function is monotone and supermodular. We propose two greedy algorithms with provable approximation guarantees. The first is a deterministic algorithm with an approximation factor (1-1/e) and O(n^3) running time; while the second is a randomized algorithm with a (1-1/e-ϵ)-approximation and O (k mϵ^-2) running time for any small ϵ>0, where the O (·) notation hides the poly factors. Extensive experiments on models 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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