Fast Estimation and Optimization of Resistance Diameter on Graphs

The resistance diameter of a graph is the maximum resistance distance among all pairs of nodes in the graph, which has found various applications in many scenarios. However, direct computation of resistance diameter involves the pseudoinverse of graph Laplacian, which takes cubic time and is thus infeasible for huge networks with millions of nodes. In this paper, we consider the computation and optimization problems for resistance diameter of a graph. First, we develop a nearly linear time algorithm to approximate the resistance diameter, which has a theoretically guaranteed error. Then, we propose and study an optimization problem of adding a fixed number of edges to a graph, such that the resistance diameter of the resulting graph is minimized. We show that the objective function is non-supermodular but monotone. Moreover, we propose two fast heuristic algorithms to approximately solve this problem. Finally, we conduct extensive experiments on different networks with sizes up to one million nodes, demonstrating the superiority of our algorithms in terms of efficiency and effectiveness.