Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs

TL;DR

This paper develops mathematical tools to analyze the large-scale structure of directed networks using maximum entropy ensembles, providing explicit formulas and applying them to gene regulation data.

Contribution

It introduces new analytical methods for directed graph ensembles with specified degree distributions and correlations, advancing network null model analysis.

Findings

Derived explicit formulas for entropy and complexity of directed graph ensembles.

Applied the methods to gene regulation networks to demonstrate practical utility.

Provided tools for quantifying topological differences between real and random networks.

Abstract

We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random graphs with prescribed joint distributions for in- and outdegrees and prescribed degree-degree correlation functions. We calculate exact and explicit formulae for the leading orders in the system size of the Shannon entropies and complexities of these ensembles, and for information-theoretic distances. The results are applied to data on gene regulation networks.