New Approximation Algorithms for Forest Closeness Centrality -- for Individual Vertices and Vertex Groups

The emergence of massive graph data sets requires fast mining algorithms. Centrality measures to identify important vertices belong to the most popular analysis methods in graph mining. A measure that is gaining attention is forest closeness centrality; it is closely related to electrical measures using current flow but can also handle disconnected graphs. Recently, [Jin et al., ICDM'19] proposed an algorithm to approximate this measure probabilistically. Their algorithm processes small inputs quickly, but does not scale well beyond hundreds of thousands of vertices. In this paper, we first propose a different approximation algorithm; it is up to two orders of magnitude faster and more accurate in practice. Our method exploits the strong connection between uniform spanning trees and forest distances by adapting and extending recent approximation algorithms for related single-vertex problems. This results in a nearly-linear time algorithm with an absolute probabilistic error guarantee. In addition, we are the first to consider the problem of finding an optimal group of vertices w.r.t. forest closeness. We prove that this latter problem is NP-hard; to approximate it, we adapt a greedy algorithm by [Li et al., WWW'19], which is based on (partial) matrix inversion. Moreover, our experiments show that on disconnected graphs, group forest closeness outperforms existing centrality measures in the context of semi-supervised vertex classification.