Approximating Kemeny's constant via spectral sparsification
Kemeny's constant of a network quantifies the expected time for a random walk to reach a target vertex chosen according to the stationary distribution, independently of the starting vertex. The purpose of this work is twofold. First, we introduce the class of slightly regular networks, whose orthogonal degree vector is an eigenvector of the Laplacian, and derive an exact expression for Kemeny's constant for this class, extending previous results beyond regular networks. Second, we address the efficient approximation of Kemeny's constant using spectral sparsification. We establish multiplicative bounds relating Kemeny's constant of a network to that of any of its spectral sparsifiers, and use these bounds to design an approximation algorithm. While standard exact approaches based on a full spectral decomposition or on the computation of a pseudoinverse have $O(n^3)$ complexity, our method has complexity $O(n^2\log^{1+o(1)}n)$ for fixed approximation parameters, and is therefore nearly quadratic in the number of vertices even for dense networks.