Kron Reduction and Effective Resistance of Directed Graphs
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In network theory, the concept of the effective resistance is a distance measure on graphs which relates global network properties to the individual connections between nodes, and the Kron reduction method is a standard tool to reduce or eliminate the desired nodes, which preserves the interconnection structure and the effective resistance of the original graph. While these two graph-theoretic concepts stem from the electric network on an undirected graph, they have a number of applications over not only that but also a wide variety of fields. In this paper, we propose a generalization of Kron reduction to directed graphs. Furthermore, we prove this reduction method preserves the structure of the original graphs such as strong connectivity or weight balancedness. Also, we generalize the effective resistance to directed graphs via Markov chain theory, which is invariant under Kron reduction. Although the effective resistance of our proposal is asymmetric, we prove that it induces the novel graph metric on strongly connected directed graphs. Finally, we compare our method with existing studies and prove that the hitting probability metric is the logarithm of the effective resistance in the stochastic case, and the effective resistance in the doubly stochastic case is the same with the resistance distance on ergodic Markov chain.
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