Power-Law Graphs Have Minimal Scaling of Kemeny Constant for Random Walks

The mean hitting time from a node i to a node j selected randomly according to the stationary distribution of random walks is called the Kemeny constant, which has found various applications. It was proved that over all graphs with N vertices, complete graphs have the exact minimum Kemeny constant, growing linearly with N. Here we study numerically or analytically the Kemeny constant on many sparse real-world and model networks with scale-free small-world topology, and show that their Kemeny constant also behaves linearly with N. Thus, sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of Kemeny constant. We then present a theoretically guaranteed estimation algorithm, which approximates the Kemeny constant for a graph in nearly linear time with respect to the number of edges. Extensive numerical experiments on model and real networks show that our approximation algorithm is both efficient and accurate.