  In network theory, the concept of effective resistance is a distance measure
on a graph that relates the global network properties to individual connections
between nodes. In addition, the Kron reduction method is a standard tool for
reducing or eliminating the desired nodes, which preserves the interconnection
structure and the effective resistance of the original graph. Although these
two graph-theoretic concepts stem from the electric network on an undirected
graph, they also have a number of applications throughout a wide variety of
other fields. In this study, we propose a generalization of a Kron reduction
for directed graphs. Furthermore, we prove that this reduction method preserves
the structure of the original graphs, such as the strong connectivity or weight
balance. In addition, we generalize the effective resistance to a directed
graph using Markov chain theory, which is invariant under a Kron reduction.
Although the effective resistance of our proposal is asymmetric, we prove that
it induces two novel graph metrics in general strongly connected directed
graphs. In particular, the effective resistance captures the commute and
covering times for strongly connected weight balanced directed graphs. Finally,
we compare our method with existing approaches and relate the hitting
probability metrics and effective resistance in a stochastic case. In addition,
we show that the effective resistance in a doubly stochastic case is the same
as the resistance distance in an ergodic Markov chain.
