Generalization of the small-world effect on a model approaching the Erd\u0151s\u2013R\xe9nyi random graph

Benjamin F Maier

doi:10.1038/s41598-019-45576-3

Benjamin F Maier

Open Access

https://doi.org/10.1038/s41598-019-45576-3

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Abstract

The famous Watts–Strogatz (WS) small-world network model does not approach the Erdős–Rényi (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.

Highlights

The famous Watts–Strogatz (WS) small-world network model does not approach the Erdős–Rényi (ER) random graph model in the limit of total randomization which can lead to confusion and complicates certain analyses
When Watts and Strogatz published their 1998 paper “Collective dynamics of ’small-world’ networks”[1], it had a phenomenal influence on the field of complex systems and was one of the defining studies for the following success of network science to emerge as an interdisciplinary field
Constructing small-world networks in this way allows for a thorough analytical analysis of network properties such as the degree distribution, the degree variance, the average number of two-stars, the average number of triangles, and the clustering coefficient

Summary

Introduction

The famous Watts–Strogatz (WS) small-world network model does not approach the Erdős–Rényi (ER) random graph model in the limit of total randomization which can lead to confusion and complicates certain analyses. A model which solves the problems discussed above has been introduced by Song and Wang[19] Within their study, they showed that sampling edges from a distance-based connection probability eases the evaluations of e.g. the degree distribution and the clustering coefficient. We reformulate and discuss this modified model, propose an efficient sampling algorithm, extend the evaluation of degree distribution and clustering coefficient to other network properties, and show how it can be used to explain the small-world effect analytically by comparing the clustering coefficient to an upper bound of the message delivery time. Since the shortest path length equals the delivery time of an optimal search process between two nodes, the result presented here generalizes the small-world effect to random search strategies

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