4 years ago

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

Benjamin F. Maier

The famous Watts-Strogatz small-world network model does not approach the Erd\H{o}s-R\'enyi random graph model in the limit of total randomization which can lead to confusion and complicates certain analyses. In this paper we revisit a simple alternative by Song and Wang, which, instead of rewiring, draws edges between pairs of nodes with a distance-based connection probability. We show that this model is simpler to analyze and approaches the true Erd\H{o}s-R\'enyi random graph model in the corresponding limit. 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. Consequently, 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. We propose this model as an alternative reference model for studying the influence of small-world topology on dynamic systems as well as a simple model to introduce numerous topics when teaching network science.

Publisher URL: http://arxiv.org/abs/1901.02381

DOI: arXiv:1901.02381v2

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