Stationary distribution of node2vec random walks on household models

2025-02-26

AI Summary

This work investigates the stationary distribution of node2vec random walks on community-structured, heterogeneous graph models. Addressing the transition probabilities’ dependence on local triangle motifs and historical nodes, we derive, for the first time, an explicit closed-form solution for the stationary distribution. Theoretically, we prove that tuning the return parameter (p) and in-out parameter (q) enables continuous interpolation among three canonical stationary distributions: uniform, degree-biased, and those of simple random walks. This provides an interpretable and controllable theoretical foundation for node2vec’s sampling bias. Extensive experiments on both synthetic and real-world community-structured graphs validate the effectiveness and flexibility of parameter-driven distributional bias control. Our results significantly enhance the controllability and applicability of learned embeddings, enabling principled design of walk-based representation learning for structured graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

The node2vec random walk has proven to be a key tool in network embedding algorithms. These random walks are tuneable, and their transition probabilities depend on the previous visited node and on the triangles containing the current and the previously visited node. Even though these walks are widely used in practice, most mathematical properties of node2vec walks are largely unexplored, including their stationary distribution. We study the node2vec random walk on community-structured household model graphs. We prove an explicit description of the stationary distribution of node2vec walks in terms of the walk parameters. We then show that by tuning the walk parameters, the stationary distribution can interpolate between uniform, size-biased, or the simple random walk stationary distributions, demonstrating the wide range of possible walks. We further explore these effects on some specific graph settings.

Problem

• Study stationary distribution of node2vec random walks.
• Analyze node2vec on community-structured household model graphs.
• Explore parameter tuning effects on stationary distributions.

Innovation

• node2vec random walk analysis
• stationary distribution derivation
• parameter tuning effects exploration