A Probabilistic Framework for Structural Analysis in Directed Networks

TL;DR

This paper extends a probabilistic framework for structural analysis from undirected to directed networks by allowing asymmetric bivariate distributions with identical marginals, enabling community detection.

Contribution

It introduces a relaxed assumption for directed networks, allowing asymmetric distributions with equal marginals, and proposes a hierarchical algorithm for community detection under this framework.

Findings

Framework extended to directed networks with asymmetric distributions.

Community detection algorithm adapted for directed networks.

Modularity remains consistent through a constructed symmetric sampled graph.

Abstract

In our recent works, we developed a probabilistic framework for structural analysis in undirected networks. The key idea of that framework is to sample a network by a symmetric bivariate distribution and then use that bivariate distribution to formerly define various notions, including centrality, relative centrality, community, and modularity. The main objective of this paper is to extend the probabilistic framework to directed networks, where the sampling bivariate distributions could be asymmetric. Our main finding is that we can relax the assumption from symmetric bivariate distributions to bivariate distributions that have the same marginal distributions. By using such a weaker assumption, we show that various notions for structural analysis in directed networks can also be defined in the same manner as before. However, since the bivariate distribution could be asymmetric, the community detection algorithms proposed in our previous work cannot be directly applied. For this, we show that one can construct another sampled graph with a symmetric bivariate distribution so that for any partition of the network, the modularity index remains the same as that of the original sampled graph. Based on this, we propose a hierarchical agglomerative algorithm that returns a partition of communities when the algorithm converges.