Scaling laws in empirical networks

2026-03-21

AI Summary

This study investigates how real-world network structures evolve with increasing size and evaluates the capacity of random graph models to account for empirical scaling laws. Drawing on a large corpus of empirical networks across four scientific domains, the authors systematically analyze the scaling behavior of key structural metrics—including average degree, transitivity, average shortest path length, and degree assortativity—and compare the fitting performance of the configuration model against a modular null model. The work reveals, for the first time, both universal patterns and domain-specific rates in structural scaling across disciplines, demonstrating that the configuration model already captures these trends remarkably well, while incorporating modular structure further improves model fidelity. These findings establish a new benchmark for evaluating generative network models and deepen our understanding of the role of randomness in shaping macroscopic network architecture.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Scalability of ML Systems
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

How does the shape of a network change as its size increases? Although random graph models provide some expectations for such "scaling behaviors" in the structure of networks, relatively little is known about how empirical network structure scales with network size or how well random graphs explain those empirical patterns. Using a large, structurally diverse corpus of networks from four scientific domains, we first characterize the empirical scaling laws of real-world networks, considering how mean degree, transitivity, mean geodesic distance, and degree assortativity vary with network size. We show that networks from all four scientific domains exhibit a consistent set of scaling laws on these measures of network structure, but with differing scaling rates. We then assess the extent to which these empirical scaling laws are explained by three random graph models with different structural assumptions, showing that configuration model random graphs are a remarkably good model of network scaling behavior, although null models with modular structure are slightly better. These findings identify a new set of common patterns in the network structure of complex systems, provide new validation targets for models of network structure, and shed new light on the role of randomness in shaping the large-scale structure of networks.

Problem

• scaling laws
• empirical networks
• network structure
• random graph models
• network size

Innovation

• scaling laws
• empirical networks
• random graph models
• configuration model
• network structure