Reproducing the first and second moment of empirical degree distributions

ArXiv ID: 2505.10373 “View on arXiv”

Authors: Mattia Marzi, Francesca Giuffrida, Diego Garlaschelli, Tiziano Squartini

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.

Keywords: Exponential Random Graphs (ERG), Network Analysis, Degree Distribution, Non-linear Models, Complex Networks, Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with advanced mathematics, including extensive use of probability theory, combinatorics, and statistical physics formalisms to derive non-linear constraints and model degeneracies. While it mentions real-world financial and social network applications and presents one illustrative figure, the focus is on theoretical derivations, approximation schemes, and model definitions rather than providing detailed empirical backtesting procedures, code, or datasets, making it more suited for theoretical exploration.

  flowchart TD
    A["Research Goal: Reproduce First & Second Moments of Empirical Degree Distributions"] --> B["Methodology: Define Fitness-Induced ERG Model"]
    A --> C["Data: Real-World Complex Network (Multi-Asset)"]
    B --> D["Computational Process: Mean-Field Approximation & Parameter Estimation"]
    C --> D
    D --> E["Outcome: Model Degenerates"]
    D --> F["Outcome: Softened Model Preserves Variance & Mean"]
    E --> G["Key Finding: Standard Mean-Field Fails for Non-Linear ERGs"]
    F --> H["Key Finding: Canonical Model Captures Degree Distribution Moments"]