Local Search for Group Closeness Maximization on Big Graphs

Abstract

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.