Random walks and diffusion on networks

TL;DR

This paper surveys the theory and diverse applications of simple random walks on networks, covering types, properties, and uses in ranking, community detection, sampling, and opinion dynamics.

Contribution

It provides a comprehensive overview of random walks on networks, highlighting their theoretical foundations and practical applications across various fields.

Findings

Different types of random walks are characterized and compared.

Applications include PageRank, community detection, and opinion models.

The survey connects theoretical concepts with real-world network analysis.

Abstract

Random walks are ubiquitous in the sciences, and they are interesting from both theoretical and practical perspectives. They are one of the most fundamental types of stochastic processes; can be used to model numerous phenomena, including diffusion, interactions, and opinions among humans and animals; and can be used to extract information about important entities or dense groups of entities in a network. Random walks have been studied for many decades on both regular lattices and (especially in the last couple of decades) on networks with a variety of structures. In the present article, we survey the theory and applications of random walks on networks, restricting ourselves to simple cases of single and non-adaptive random walkers. We distinguish three main types of random walks: discrete-time random walks, node-centric continuous-time random walks, and edge-centric continuous-time random walks. We first briefly survey random walks on a line, and then we consider random walks on various types of networks. We extensively discuss applications of random walks, including ranking of nodes (e.g., PageRank), community detection, respondent-driven sampling, and opinion models such as voter models.