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October 2008 Neighbor selection and hitting probability in small-world graphs
Oskar Sandberg
Ann. Appl. Probab. 18(5): 1771-1793 (October 2008). DOI: 10.1214/07-AAP499

Abstract

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself.

In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.

Citation

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Oskar Sandberg. "Neighbor selection and hitting probability in small-world graphs." Ann. Appl. Probab. 18 (5) 1771 - 1793, October 2008. https://doi.org/10.1214/07-AAP499

Information

Published: October 2008
First available in Project Euclid: 30 October 2008

zbMATH: 1152.60014
MathSciNet: MR2462548
Digital Object Identifier: 10.1214/07-AAP499

Subjects:
Primary: 60C05
Secondary: 68R10, 68W20

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.18 • No. 5 • October 2008
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