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arXiv:1901.02381 (physics)
[Submitted on 8 Jan 2019 (v1), last revised 25 Jun 2019 (this version, v3)]

Title:Generalization of the small-world effect on a model approaching the Erdős-Rényi random graph

Authors:Benjamin F. Maier
View a PDF of the paper titled Generalization of the small-world effect on a model approaching the Erd\H{o}s-R\'enyi random graph, by Benjamin F. Maier
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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.
Comments: 6 pages, 5 figures
Subjects: Physics and Society (physics.soc-ph); Social and Information Networks (cs.SI)
Cite as: arXiv:1901.02381 [physics.soc-ph]
  (or arXiv:1901.02381v3 [physics.soc-ph] for this version)
  https://doi.org/10.48550/arXiv.1901.02381
arXiv-issued DOI via DataCite
Journal reference: Scientific Reports (2019) 9:9268
Related DOI: https://doi.org/10.1038/s41598-019-45576-3
DOI(s) linking to related resources

Submission history

From: Benjamin F. Maier [view email]
[v1] Tue, 8 Jan 2019 16:02:07 UTC (132 KB)
[v2] Wed, 9 Jan 2019 13:59:53 UTC (132 KB)
[v3] Tue, 25 Jun 2019 10:56:29 UTC (356 KB)
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