Random walks on weighted networks: a survey of local and non-local dynamics

Abstract

Abstract In this article, we present a survey of different types of random walk models with local and non-local transitions on undirected weighted networks. We present a general approach by defining the dynamics as a discrete-time Markovian process with transition probabilities expressed in terms of a symmetric matrix of weights. In the first part, we describe the matrices of weights that define local random walk dynamics like the normal random walk, biased random walks and preferential navigation, random walks in the context of digital image processing and maximum entropy random walks. In addition, we explore non-local random walks, like Lévy flights on networks, fractional transport through the new formalism of fractional graph Laplacians, 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 random walks 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 dynamics, and we discuss their capacity to explore several types of networks. Our results allow us to study and compare the global dynamics of different types of random walk models.