Repelling Random Walks

TL;DR

This work addresses the challenge of efficiently sampling on graphs without bias by introducing repelling random walks, a discrete quasi-Monte Carlo scheme that induces correlations among multiple walkers while preserving each walk's marginal transition probabilities. The method yields unbiased estimators with strengthened concentration and is simple to implement as a drop-in modification (sampling without replacement within neighbor blocks). The authors develop theory and demonstrate strong gains across three domains: graph-kernel estimation via Graph Random Features, PageRank approximation, and graphlet concentration estimation, including a theoretical variance-reduction result for kernels and a general PageRank variance bound. Empirically, repelling walkers consistently outperform iid walkers across synthetic and real graphs, suggesting broad practical impact for graph-based statistical estimation and learning tasks. The work opens avenues for further theoretical analysis and extensions of quasi-Monte Carlo sampling to a wider class of graph estimators and networks.

Abstract

We present a novel quasi-Monte Carlo mechanism to improve graph-based sampling, coined repelling random walks. By inducing correlations between the trajectories of an interacting ensemble such that their marginal transition probabilities are unmodified, we are able to explore the graph more efficiently, improving the concentration of statistical estimators whilst leaving them unbiased. The mechanism has a trivial drop-in implementation. We showcase the effectiveness of repelling random walks in a range of settings including estimation of graph kernels, the PageRank vector and graphlet concentrations. We provide detailed experimental evaluation and robust theoretical guarantees. To our knowledge, repelling random walks constitute the first rigorously studied quasi-Monte Carlo scheme correlating the directions of walkers on a graph, inviting new research in this exciting nascent domain.

Figures (4)

• Figure 1: Schematic for behaviour of repelling random walkers at a particular timestep. By sampling from each 'block' (blue and green rectangles) without replacement we get a more even distribution over neighbours, without changing the marginal probabilities.
• Figure 2: Relative Frobenius norm of estimates of the $2$-regularised Laplace kernel (lower is better) vs. number of random walks for: i) vanilla GRFs; ii) GRFs with antithetic termination reid2023quasi ('q-a-GRFs'); iii) GRFs with repelling walks ('q-r-GRFs'); iv) GRFs with both antithetic termination and repelling walks ('q-ar-GRFs'). Using both QMC schemes together gives the best results for all graphs considered and the gains are large (sometimes a factor of $>2$). $N$ gives the number of nodes, $p$ is the edge-generation probability for the Erdös-Rényi graphs, and $d$ is the $d$-regular node degree. One standard deviation on the mean error is shaded but is too small to easily see.
• Figure 3: Graphlets for $k=3$
• Figure 4: Mean square error on estimates of $k=3$ graphlet concentrations with different numbers of random walks on different graphs. Lower is better. Using the repelling scheme consistently improves the quality of the estimate compared to independent walks.

Theorems & Definitions (6)

• Definition 2.1: Repelling random walks
• Definition 2.2: Transient repelling random walks
• Theorem 3.1: Superiority of repelling random walks for kernel estimation
• Theorem 4.2: Superiority of repelling random walks for PageRank estimation
• Definition 4.3: Step-by-step linear functions
• Theorem 4.4: Variance of step-by-step linear functions is reduced by transient repulsion