Reproducing the first and second moments of empirical degree distributions

TL;DR

This paper introduces a non-linear exponential random graph model that accurately reproduces both the mean and variance of empirical degree distributions in complex networks, overcoming limitations of linear models.

Contribution

It develops a fitness-induced non-linear ERG model that captures degree distribution variance, extending the capabilities of traditional linear ERGs.

Findings

The mean-field approximation causes degeneracy in the degree-corrected two-star model.

The proposed softened model reproduces empirical degree variance.

The model retains explanatory power within a canonical framework.

Abstract

The study of probabilistic models for the analysis of complex networks represents a flourishing research field. Among the former, Exponential Random Graphs (ERGs) have gained increasing attention over the years. So far, only linear ERGs have been extensively employed to gain insight into the structural organisation of real-world complex networks. None, however, is capable of accounting for the variance of the empirical degree distribution. To this aim, non-linear ERGs must be considered. After showing that the usual mean-field approximation forces the degree-corrected version of the two-star model to degenerate, we define a fitness-induced variant of it. Such a `softened' model is capable of reproducing the sample variance, while retaining the explanatory power of its linear counterpart, within a purely canonical framework.