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Phase transition in random distance graphs on the torus

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  30 November 2017

Fioralba Ajazi ,
George M. Napolitano  and
Tatyana Turova
Fioralba Ajazi*
Affiliation:
Lund University and University of Lausanne
George M. Napolitano*
Affiliation:
Lund University
Tatyana Turova*
Affiliation:
Lund University
*
* Postal address: Department of Mathematical Statistics, Faculty of Science, Lund University, Sölvegatan 18, 22100 Lund, Sweden.
** Current address: Centre for Epidemiology and Screening, Department of Public Health, University of Copenhagen, Øster Farimagsgade 5, 1014 Copenhagen, Denmark.
* Postal address: Department of Mathematical Statistics, Faculty of Science, Lund University, Sölvegatan 18, 22100 Lund, Sweden.
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Abstract

In this paper we consider random distance graphs motivated by applications in neurobiology. These models can be viewed as examples of inhomogeneous random graphs, notably outside of the so-called rank-1 case. Treating these models in the context of the general theory of inhomogeneous graphs helps us to derive the asymptotics for the size of the largest connected component. In particular, we show that certain random distance graphs behave exactly as the classical Erdős–Rényi model, not only in the supercritical phase (as already known) but in the subcritical case as well.

Keywords

Inhomogeneous random graphrandom distance graphlargest connected component

MSC classification

Primary: 05C80: Random graphs 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1278 - 1294
Copyright
Copyright © Applied Probability Trust 2017 

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