Exact results and scaling properties of small-world networks

TL;DR

This paper analytically investigates the distribution and scaling properties of minimal paths in small-world networks, providing simplified calculations for average distances and their variances, especially in large systems.

Contribution

It derives exact analytic results for the distribution of minimal paths and their scaling behavior in small-world networks, advancing understanding beyond numerical methods.

Findings

Analytic expressions for the distribution function of minimal paths.

Simplified calculation methods for average minimal distance and variance.

Scaling laws for large system sizes in small-world networks.

Abstract

We study the distribution function for minimal paths in small-world networks. Using properties of this distribution function, we derive analytic results which greatly simplify the numerical calculation of the average minimal distance, $\bar{\ell}$, and its variance, $\sigma^2$. We also discuss the scaling properties of the distribution function. Finally, we study the limit of large system sizes and obtain some analytic results.