node2vec or triangle-biased random walks: stationarity, regularity & recurrence

2026-04-15

AI Summary

This study investigates the long-term behavior of the node2vec random walk—a non-Markovian process—on arbitrary graphs, focusing on its stationarity, ergodicity, reversibility, and recurrence. By lifting the walk to spaces of directed edges and directed wedges, the authors construct two higher-order Markov representations and employ tools from ergodic theory and graph theory to analyze its asymptotic properties. The main contributions include establishing sufficient conditions under which these stochastic properties hold, demonstrating that on regular graphs the wedge-based representation simplifies the node2vec dynamics, and proving an equivalence between graph regularity and weighted Eulerianicity. Furthermore, the work clarifies the fundamental distinction between node2vec and non-backtracking random walks.

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: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The node2vec random walk is a non-Markovian random walk on the vertex set of a graph, widely used for network embedding and exploration. This random walk model is defined in terms of three parameters which control the probability of, respectively, backtracking moves, moves within triangles, and moves to the remaining neighboring nodes. From a mathematical standpoint, the node2vec random walk is a nontrivial generalization of the non-backtracking random walk and thus belongs to the class of second-order Markov chains. Despite its widespread use in applications, little is known about its long-run behavior. The goal of this paper is to begin exploring its fundamental properties on arbitrary graphs. To this aim, we show how lifting the node2vec random walk to the state spaces of directed edges and directed wedges yields two distinct Markovian representations which are key for its asymptotic analysis. Using these representations, we find mild sufficient conditions on the underlying finite or infinite graph to guarantee ergodicity, reversibility, recurrence and characterization of the invariant measure. As we discuss, the behavior of the node2vec random walk is drastically different compared to the non-backtracking random walk. While the latter simplifies on arbitrary graphs when using its natural edge Markovian representation thanks to bistochasticity, the former simplifies on regular graphs when using its natural wedge Markovian representation. Remarkably, this representation reveals that a graph is regular if and only if a certain weighted Eulerianity condition holds.

Problem

• node2vec
• random walk
• stationarity
• recurrence
• regularity

Innovation

• node2vec
• non-Markovian random walk
• Markovian representation
• ergodicity
• regular graph