[Paper Review] Random walks on weighted networks: Exploring local and non-local navigation strategies

Alejandro P. Riascos, José L. Mateos|arXiv (Cornell University)|Jan 17, 2019

Diffusion and Search Dynamics10 references4 citations

TL;DR

This paper presents a unified framework for analyzing local and non-local random walk strategies on weighted networks using a symmetric matrix of weights to define transition probabilities. It derives exact expressions for key dynamical quantities—stationary distribution, mean first passage time, Kemeny’s constant, and global exploration time—using the eigenvalues and eigenvectors of the transition matrix, demonstrating that non-local strategies like Lévy flights and fractional transport significantly enhance network exploration efficiency compared to local walks.

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 Lévy 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.

Motivation & Objective

• To develop a general mathematical framework for analyzing both local and non-local random walk strategies on undirected weighted networks.
• To unify diverse random walk models—such as biased walks, Lévy flights, and fractional transport—under a single formalism based on symmetric weight matrices.
• To derive analytical expressions for fundamental dynamical quantities like stationary distribution, mean first passage time, and global exploration time.
• To compare the efficiency of different random walk strategies in exploring network structures using spectral properties of the transition matrix.
• To apply the formalism to human mobility models, showing how hybrid local-non-local strategies improve site visitation efficiency in spatial regions.

Proposed method

• Model random walks as discrete-time Markov processes with transition probabilities derived from a symmetric matrix of weights.
• Define the transition matrix Π using the weight matrix elements, ensuring detailed balance and symmetric stationary distributions.
• Use spectral decomposition of the transition matrix to compute the stationary distribution, mean first passage time, and Kemeny’s constant.
• Express global exploration time τ and Kemeny’s constant in terms of eigenvalues and eigenvectors of the transition matrix.
• Apply the formalism to specific strategies: preferential random walks, Lévy flights on networks, and fractional transport via the fractional Laplacian.
• Numerically evaluate eigenvalues and eigenvectors for real-world and synthetic networks to compute τ(α,R) for human mobility applications.

Experimental results

Research questions

• RQ1 How can local and non-local random walk strategies on weighted networks be systematically compared using a unified mathematical framework?
• RQ2 What role do the eigenvalues and eigenvectors of the transition matrix play in determining the global exploration dynamics of random walks?
• RQ3 How does the global exploration time τ vary across different random walk strategies, and what does it reveal about their efficiency?
• RQ4 To what extent do non-local strategies like Lévy flights and fractional transport outperform local random walks in visiting nodes or target locations?
• RQ5 Can the formalism be extended to spatial mobility models where nodes are not predefined but locations are visited in continuous space?

Key findings

• The global exploration time τ is a robust quantitative measure for comparing the efficiency of different random walk strategies, with lower τ indicating faster coverage of target nodes.
• For Lévy flight and fractional transport strategies, τ remains nearly constant across different neighborhood radii R when α ≤ 1, indicating optimal non-local exploration independent of local scale.
• When α ≫ 1, the strategy becomes local and τ increases significantly, showing that long-range jumps are essential for efficient exploration.
• The Kemeny’s constant, derived from the spectrum of the transition matrix, provides a simplified yet accurate measure of exploration efficiency for strategies with uniform stationary distributions.
• In human mobility applications, hybrid local-non-local strategies defined by Ω_ij^(α)(R) achieve minimal τ for α ≤ 1, confirming that non-locality enhances site visitation efficiency.
• The analytical expressions for τ(α,R) are validated numerically using exact eigenvalue and eigenvector computations, confirming the theoretical predictions.