Random walks on weighted networks: Exploring local and non-local navigation strategies

Abstract

In this paper, we present an overview of different types of random walk strategies with local and non-local transitions on undirected connected networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete time Markovian process with probabilities of transition expressed in terms of a symmetric matrix of weights. In the first part, we describe the matrices of weights that define local random walk strategies like the normal random walk, biased random walks, random walks in the context of digital image processing and maximum entropy random walks. In addition, we explore non-local random walks like L\'evy flights on networks, fractional transport and applications in the context of human mobility. Explicit relations for the stationary probability distribution, the mean first passage time and global times to characterize the random walk strategies are obtained in terms of the elements of the matrix of weights and its respective eigenvalues and eigenvectors. Finally, we apply the results to the analysis of particular local and non-local random walk strategies; we discuss their efficiency and capacity to explore different types of structures. Our results allow to study and compare on the same basis the global dynamics of different types of random walk strategies.