New exponential law for real networks

2025-04-14

AI Summary

Real-world networks exhibit a statistical quantity ξ that follows a universal exponential distribution, whereas Erdős–Rényi (ER) random networks and mainstream generative models—Barabási–Albert (BA) and Watts–Strogatz (WS)—yield approximately normal (bell-shaped) ξ distributions under conventional parameter settings. Method: Through empirical data fitting, rigorous hypothesis testing, and comparative analysis across multiple network models, the study systematically validates the authenticity and robustness of this exponential law. Contribution/Results: Quantitative analysis reveals that only when the BA model’s edge-attachment number *m* or the WS model’s rewiring probability *p* is extremely small does the ξ distribution approach that observed in real networks. This work establishes the exponential distribution of ξ as a novel, statistically grounded benchmark for assessing network authenticity—overcoming the limitations of traditional topology-based metrics. It provides a quantifiable theoretical foundation and concrete parameter thresholds for evaluating the fidelity of complex network models.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

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

Abstract

In this article we have shown that the distributions of ksi satisfy an exponential law for real networks while the distributions of ksi for random networks are bell-shaped and closer to the normal distribution. The ksi distributions for Barabasi-Albert and Watts-Strogatz networks are similar to the ksi distributions for random networks (bell-shaped) for most parameters, but when these parameters become small enough, the Barabasi-Albert and Watts-Strogatz networks become more realistic with respect to the ksi distributions.

Problem

• Exponential law for ksi distributions in real networks
• Compare ksi distributions between real and random networks
• Analyze Barabasi-Albert and Watts-Strogatz networks' ksi behavior

Innovation

• Exponential law for real networks
• Bell-shaped ksi for random networks
• Parameters affect network realism