Efficient Approximation of Kemeny's Constant for Large Graphs

Abstract

For an undirected graph, its Kemeny's constant is defined as the mean hitting time of random walks from one vertex to another chosen randomly according to the stationary distribution. Kemeny's constant exhibits numerous explanations from different perspectives and has found various applications in the field of complex networks. Due to the requirement of computing the inverse of the normalized Laplacian matrix, it is infeasible to get the accurate Kemeny's constant of large networks with millions of vertices. Existing methods either consume excessive memory space that are impractical for large-scale networks, or involve redundant simulation, leaving room for further optimization. In this paper, we propose two scalable Monte Carlo algorithms RefinedMC and ForestMC to approximate Kemeny's constant. RefinedMC makes several refinements based on the simulation of truncated random walks, significantly reducing the amount of required random walks, while ForestMC utilizes the newly discovered paradigm connecting Kemeny's constant with the inverse of corresponding Laplacian submatrix, which is considerably accurate. Extensive numerical experiments on model and realistic networks demonstrate that our approximation algorithms evidently outperform the baseline methods in terms of efficiency and accuracy.