Analysis of node2vec random walks on networks.

Lingqi Meng,Naoki Masuda

doi:10.1098/rspa.2020.0447

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 analyse 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 analyse 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 backtracking and visiting a neighbour of the previously visited node but do not avoid them completely.

Highlights

Random walks on finite networks have been a favourite research topic for decades [1,2,3,4]
We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyse the stationary probability, relaxation times in terms of the spectral gap of the transition probability matrix, and coalescence time
We show that node2vec random walk accelerates diffusion when walkers are designed to avoid both backtracking and visiting a neighbour of the previously visited node but do not avoid them completely

Summary

Introduction

Random walks on finite networks have been a favourite research topic for decades [1,2,3,4]. We provide multiple lines of evidence supporting that diffusion (i.e. approaching to the stationary probability and coalescence of random walkers) is accelerated when the parameters of node2vec random walks are tuned such that backtracking and visiting the neighbours of the last visited node are suppressed and exploration of the rest of the network, similar to depth-first sampling, is explicitly promoted. This is the case unless the avoidance of local sampling including backtracking is not excessive.

Results

Discussion