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This paper analyzes the long-run behavior of the node2vec random walk, a widely used non-Markovian random walk for network embedding. By lifting the node2vec random walk to Markovian representations on directed edges and wedges, the authors derive sufficient conditions for ergodicity, reversibility, and recurrence on arbitrary graphs. A key finding is that node2vec simplifies on regular graphs via its wedge Markovian representation, unlike the non-backtracking random walk which simplifies on arbitrary graphs.
Node2vec's behavior diverges sharply from simpler random walks, simplifying on regular graphs through a novel wedge-based representation, revealing a surprising link between graph regularity and a weighted Eulerianity condition.
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.