# Generalization of the small-world effect on a model approaching the Erd\H{o}s-R\'enyi random graph

## TL;DR

This paper introduces a simple, analyzable alternative to the Watts-Strogatz small-world model that approaches the Erdős-Rényi random graph in the limit, providing clearer insights into small-world phenomena and their effects.

## Contribution

The paper presents a distance-based connection probability model that simplifies analysis, approaches ER graphs in the limit, and demonstrates differences from WS models in network dynamics.

## Key findings

- The alternative model approaches ER graphs in the fully randomized limit.
- Clustering coefficient decreases slower than message delivery time upper bound.
- The model enables analytic evaluation of degree distribution, clustering, and random walk properties.

## Abstract

The famous Watts-Strogatz (WS) small-world network model does not approach the Erd\H{o}s-R\'enyi (ER) random graph model in the limit of total randomization which can lead to confusion and complicates certain analyses. In this paper we discuss a simple alternative which was first introduced by Song and Wang, where instead of rewiring, edges are drawn between pairs of nodes with a distance-based connection probability. We show that this model is simpler to analyze, approaches the true ER random graph model in the completely randomized limit, and demonstrate that the WS model and the alternative model may yield different quantitative results using the example of a random walk temporal observable. An efficient sampling algorithm for the alternative model is proposed. Analytic results regarding the degree distribution, degree variance, number of two-stars per node, number of triangles per node, clustering coefficient, and random walk mixing time are presented. Subsequently, the small-world effect is illustrated by showing that the clustering coefficient decreases much slower than an upper bound on the message delivery time with increasing long-range connection probability which generalizes the small-world effect from informed searches to random search strategies. Due to its accessibility for analytic evaluations, we propose that this modified model should be used as an alternative reference model for studying the influence of small-world topologies on dynamic systems as well as a simple model to introduce numerous topics when teaching network science.