Exactly solvable analogy of small-world networks

TL;DR

This paper provides an exact analytical description of the transition in average shortest path length in a simplified small-world network model, bridging ordered and random regimes with explicit formulas.

Contribution

It introduces an exactly solvable model that captures the crossover behavior of small-world networks, offering analytical insights into their structural properties.

Findings

Derived the distribution of shortest distances $P(\, ext{ell})$

Obtained a scaling form for the average shortest path $ar{ ext{ell}}(p,n)$

Results closely match numerical data for typical small-world networks

Abstract

We present an exact description of a crossover between two different regimes of simple analogies of small-world networks. Each of the sites chosen with a probability $p$ from $n$ sites of an ordered system defined on a circle is connected to all other sites selected in such a way. Every link is of a unit length. Thus, while $p$ changes from 0 to 1, an averaged shortest distance between a pair of sites changes from $\bar{\ell} \sim n$ to $\bar{\ell} = 1$. We find the distribution of the shortest distances $P(\ell)$ and obtain a scaling form of $\bar{\ell}(p,n)$. In spite of the simplicity of the models under consideration, the results appear to be surprisingly close to those obtained numerically for usual small-world networks.