Analysis of node2vec random walks on networks

TL;DR

This paper provides a theoretical and numerical analysis of node2vec's biased random walks, revealing how specific walk strategies influence diffusion and embedding quality in network analysis.

Contribution

It offers a novel theoretical framework for understanding node2vec's second-order Markov chain walks and their impact on network diffusion and embedding performance.

Findings

Node2vec random walks accelerate diffusion when avoiding back-tracking and neighboring revisits.

Spectral gap analysis relates relaxation times to transition matrix properties.

Walk strategies influence embedding effectiveness and network exploration.

Abstract

Random walks have been proven to be useful for constructing various algorithms to gain information on networks. Algorithm node2vec employs biased random walks to realize embeddings of nodes into low-dimensional spaces, which can then be used for tasks such as multi-label classification and link prediction. The performance of the node2vec algorithm in these applications is considered to depend on properties of random walks that the algorithm uses. In the present study, we theoretically and numerically analyze random walks used by the node2vec. Those random walks are second-order Markov chains. We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyze the stationary probability, relaxation times in terms of the spectral gap of the transition probability matrix, and coalescence time. In particular, we show that node2vec random walk accelerates diffusion when walkers are designed to avoid both back-tracking and visiting a neighbor of the previously visited node but do not avoid them completely.