Reproducing the first and second moment of empirical degree distributions

2025-05-15

AI Summary

Traditional linear exponential random graph models (ERGMs) struggle to simultaneously capture both the mean and variance of empirical network degree distributions; moreover, mean-field approximations often induce model degeneracy and impair interpretability. To address this, we propose a nonlinear ERGM framework featuring a softened two-star model grounded in node fitness. For the first time under the pure canonical ensemble, our approach achieves exact joint matching of the degree distribution’s first moment (mean) and second moment (variance). By eschewing mean-field approximations, it preserves the parameter interpretability inherent in linear ERGMs while substantially enhancing statistical fidelity. Experiments demonstrate that the model robustly reproduces degree distribution characteristics across diverse real-world complex networks, effectively overcoming the degeneracy issues plaguing classical two-star models. This work establishes a new modeling paradigm for network structure—one that reconciles theoretical rigor with empirical consistency.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

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.

Problem

• Reproducing empirical degree distribution moments
• Addressing variance limitation in linear ERGs
• Developing a fitness-induced non-linear ERG model

Innovation

• Non-linear ERGs reproduce degree distribution variance
• Fitness-induced variant avoids model degeneration
• Softened model retains linear explanatory power