Mean-field solution of the small-world network model

TL;DR

This paper provides an exact mean-field solution for the average and distribution of path lengths in the small-world network model, which combines regular lattices and random shortcuts, relevant for social network analysis.

Contribution

It introduces a mean-field analytical approach to determine path length distributions in the small-world network model, valid for large system sizes.

Findings

Exact mean-field solution for average path length

Distribution of path lengths derived analytically

Applicable in the limit of large system size and shortcut density

Abstract

The small-world network model is a simple model of the structure of social networks, which simultaneously possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either large or small number of shortcuts.