Universal power laws in the threshold network model: A theoretical analysis based on extreme value theory

TL;DR

This paper provides a theoretical analysis of the threshold network model using extreme value theory, revealing universal power law behaviors in degree distribution and clustering coefficient.

Contribution

It introduces a novel theoretical framework based on extreme value theory to derive exact expressions for network properties in the threshold model.

Findings

Degree distribution follows a power law within certain ranges.

Clustering coefficient exhibits power law behavior.

Theoretical results match numerical simulations closely.

Abstract

We theoretically and numerically investigated the threshold network model with a generic weight function where there were a large number of nodes and a high threshold. Our analysis was based on extreme value theory, which gave us a theoretical understanding of the distribution of independent and identically distributed random variables within a sufficiently high range. Specifically, the distribution could be generally expressed by a generalized Pareto distribution, which enabled us to formulate the generic weight distribution function. By using the theorem, we obtained the exact expressions of degree distribution and clustering coefficient which behaved as universal power laws within certain ranges of degrees. We also compared the theoretical predictions with numerical results and found that they were extremely consistent.