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.

翻译：过去几年中，电阻距离得到了广泛研究，但大多数先前研究集中于无向网络，尽管许多现实网络是有向的。尽管已有几种将电阻距离推广至有向图的方法被提出，但它们要么缺乏物理解释，要么不是度量。本文首先基于随机游走将电阻距离的定义扩展到强连通有向图，并证明有向图中两点间的电阻距离是一个度量。随后，我们引入有向图的拉普拉斯矩阵（该矩阵包含无向图拉普拉斯矩阵作为特例），并利用其伪逆表达两点电阻距离以及由电阻距离导出的许多其他相关量。此外，我们定义了有向图中顶点与顶点组之间的电阻距离，并进一步定义了最优选择固定数量节点组以最小化其电阻距离的问题。由于该组合优化问题是NP难的，我们提出了一种具有可证明近似比的贪心算法，并在模型网络和现实网络上进行实验以验证该近似算法的性能。