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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 ofvertices; for popular measures it was shown that the problem of finding themost central group is $\mathcal{NP}$-hard. As a result, approximationalgorithms to maximize group centralities were introduced recently. Despite anearly-linear running time, approximation algorithms for group betweenness and(to a lesser extent) group closeness are rather slow on large networks due tohigh constant overheads.  That is why we introduce GED-Walk centrality, a new submodular groupcentrality measure inspired by Katz centrality. In contrast to closeness andbetweenness, it considers walks of any length rather than shortest paths, withshorter 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 $\mathcal{NP}$-hard) problem offinding a group with highest GED-Walk score.  Experiments on several real-world datasets show that scores obtained byGED-Walk improve performance on common graph mining tasks such as collectiveclassification and graph-level classification. An evaluation of empiricalrunning times demonstrates that maximizing GED-Walk (in approximation) is twoorders of magnitude faster compared to group betweenness approximation and forgroup sizes $\leq 100$ one to two orders faster than group closenessapproximation. For graphs with tens of millions of edges, approximate GED-Walkmaximization typically needs less than one minute. Furthermore, our experimentssuggest that the maximization algorithms scale linearly with the size of theinput graph and the size of the group.

Group Centrality Maximization for Large-scale Graphs