Scaling up Group Closeness Maximization

Closeness is a widely-used centrality measure in social network analysis. For a node it indicates the reciprocal of the 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 has been proposed [Chen et al., ADC 2016]. The approximation factor of (1 - 1/e) proposed by Chen et al. for this algorithm does not hold, though, as we show in this version of our paper. Since their implementation of the greedy algorithm was still too slow for large networks, Chen et al. also proposed a heuristic without approximation guarantee. In the present paper we develop new techniques to speed up the greedy algorithm. Compared to the previous 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 estimate the group with maximum closeness on networks with up to hundreds of millions of edges in minutes or at most a few hours. The greedy approach by [Chen et al., ADC 2016] would take several days already on networks with hundreds of thousands of edges. Our experiments show that the solution found by Greedy++ is actually very close to the optimum (...) Note: This paper version fixes the issue of relying on the presumed (but incorrect) submodularity of group closeness. While this has implications on the theoretical assessment of the greedy algorithm, our algorithm variant and its implementation remain unaffected. The reason is that Greedy++ relies (among others) on the supermodularity of farness, which does hold.