On role extraction for digraphs via neighbourhood pattern similarity

TL;DR

This paper analyzes how neighbourhood pattern similarity can be used to accurately recover roles in directed networks modeled by stochastic block models, with theoretical guarantees and practical experiments.

Contribution

It provides a theoretical analysis using random matrix theory showing asymptotic correctness of role recovery via neighbourhood pattern similarity in directed stochastic block models.

Findings

Asymptotic correctness of role recovery when graph size is large.

A spectral gap guarantees identification of roles.

Numerical experiments confirm effectiveness on finite networks.

Abstract

We analyse the recovery of different roles in a network modelled by a directed graph, based on the so-called Neighbourhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that when assuming the graph is generated as a particular Stochastic Block Model with Bernoulli probability distributions for the different blocks, then the recovery is asymptotically correct when the graph has a sufficiently large dimension. Under these assumptions there is a sufficient gap between the dominant and dominated eigenvalues of the similarity matrix, which guarantees the asymptotic correct identification of the number of different roles. We also comment on the connections with the literature on Stochastic Block Models, including the case of probabilities of order log(n)/n where n is the graph size. We provide numerical experiments to assess the effectiveness of the method when applied to practical networks of finite size.