Current Flow Group Closeness Centrality for Complex Networks

TL;DR

This paper extends current flow closeness centrality to groups of vertices in weighted networks, formulates the problem of maximizing group CFCC, proves NP-hardness, and proposes efficient greedy algorithms with theoretical guarantees, validated by extensive experiments.

Contribution

It introduces a novel group CFCC measure, proves the NP-hardness of maximizing it, and develops two scalable greedy algorithms with provable approximation guarantees.

Findings

The problem of maximizing group CFCC is NP-hard.

The proposed algorithms achieve near-optimal solutions with theoretical guarantees.

Algorithms are effective and scalable, handling networks with over a million vertices.

Abstract

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 $(1-\frac{k}{k-1}\cdot\frac{1}{e})$ and cubic running time; while the second is a randomized algorithm with a $(1-\frac{k}{k-1}\cdot\frac{1}{e}-\epsilon)$-approximation and nearly-linear running time for any $\epsilon > 0$. 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.