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Ultra-small scale-free geometric networks

Part of: Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  14 July 2016

J. E. Yukich
J. E. Yukich*
Affiliation:
Lehigh University
*
∗Postal address: Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. Email address: [email protected]
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Abstract

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We consider a family of long-range percolation models (Gp)p>0 on ℤd that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in Gp has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of logpdlogpd|x - y| with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from Gp(ℤd ∩ [0, n]d), resulting in two large, disconnected subgraphs.

Keywords

Scale-free graphlong-range percolationchemical distance

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 665 - 677
Copyright
© Applied Probability Trust 2006 

Footnotes

Partially supported by NSF grant DMS-0203720.

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