A simple, distance-dependent formulation of the Watts-Strogatz model for directed and undirected small-world networks

TL;DR

This paper introduces a simplified, distance-dependent formulation of the Watts-Strogatz small-world network model, enabling exact calculations of network properties for both directed and undirected networks.

Contribution

It presents an alternative, more intuitive formulation of the WS model that allows for analytical derivations of key network metrics.

Findings

Exact expressions for degree distributions

Exact motif distributions

Exact global clustering coefficient

Abstract

Small-world networks---complex networks characterized by a combination of high clustering and short path lengths---are widely studied using the paradigmatic model of Watts and Strogatz (WS). Although the WS model is already quite minimal and intuitive, we describe an alternative formulation of the WS model in terms of a distance-dependent probability of connection that further simplifies, both practically and theoretically, the generation of directed and undirected WS-type small-world networks. In addition to highlighting an essential feature of the WS model that has previously been overlooked, this alternative formulation makes it possible to derive exact expressions for quantities such as the degree and motif distributions and global clustering coefficient for both directed and undirected networks in terms of model parameters.