Random walks on weighted networks

TL;DR

This paper analytically investigates how weights in networks influence random walk dynamics, deriving key formulas for stationary distribution and mean first-passage times, with implications for controlling processes on complex networks.

Contribution

It introduces a spectral graph theory approach to analyze weighted networks, providing explicit formulas for random walk metrics and exploring effects in uncorrelated scale-free networks.

Findings

Weights significantly affect stationary distribution and MFPT.

Explicit formulas for uncorrelated networks are derived.

ATT scaling varies with network size and weight parameter.

Abstract

Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two nodes has a tunable parameter. By using the spectral graph theory, we derive analytical expressions for the stationary distribution, mean first-passage time (MFPT), average trapping time (ATT), and lower bound of the ATT, which is defined as the average MFPT to a given node over every starting point chosen from the stationary distribution. All these results depend on the weight parameter, indicating a significant role of network weights on random walks. For the case of uncorrelated networks, we provide explicit formulas for the stationary distribution as well as ATT. Particularly, for uncorrelated scale-free networks, when the target is placed on a node with the highest degree, we show that ATT can display various scalings of network size, depending also on the same parameter. Our findings could pave a way to delicately controlling random-walk dynamics on complex networks.