Mean Commute Time for Random Walks on Hierarchical Scale-Free Networks

Abstract

In recent years, there has been a surge of research interest in networks with scale-free topologies, partly due to the fact that they are prevalent in scientific research and real-life applications. In this paper, we study random-walk issues on a family of two-parameter scale-free networks, called $(x, y)$-flowers. These networks, which are constructed in a deterministic recursive fashion, display rich behaviors such as the small-world phenomenon and pseudofractal properties. We derive analytically the mean commute times for random walks on $(x, y)$-flowers and show that the mean commute times scale with the network size as a power-law function with exponent governed by both parameters $x$ and $y$. We also determine the mean effective resistance and demonstrate that it changes sharply between different choices of $x$ and $y$. Furthermore, we compare mean commute times for $(x, y)$-flowers with those for Erdős–Rényi random graphs. Our theoretical results are verified by numerical studies.