  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.
