  Resistance distance has been studied extensively in the past years, with the
majority of previous studies devoted to undirected networks, in spite of the
fact that various realistic networks are directed. Although several
generalizations of resistance distance on directed graphs have been proposed,
they either have no physical interpretation or are not a metric. In this paper,
we first extend the definition of resistance distance to strongly connected
directed graphs based on random walks and show that the two-node resistance
distance on directed graphs is a metric. Then, we introduce the Laplacian
matrix for directed graphs that subsumes the Laplacian matrix of undirected
graphs as a particular case and use its pseudoinverse to express the two-node
resistance distance, and many other relevant quantities derived from resistance
distances. Moreover, we define the resistance distance between a vertex and a
vertex group on directed graphs and further define a problem of optimally
selecting a group of fixed number of nodes, such that their resistance distance
is minimized. Since this combinatorial optimization problem is NP-hard, we
present a greedy algorithm with a proved approximation ratio, and conduct
experiments on model and realistic networks to validate the performance of this
approximation algorithm.
