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Contiguity and remote contiguity of some random graphs

Part of: Stochastic processes Probability theory on algebraic and topological structures Graph theory

Published online by Cambridge University Press:  19 August 2025

Bas J. K. Kleijn  and
Stefano Rizzelli Open the ORCID record for Stefano Rizzelli [Opens in a new window]
Bas J. K. Kleijn*
Affiliation:
University of Amsterdam
Stefano Rizzelli*
Affiliation:
University of Padova
*
*Postal address: Korteweg-de Vries Institute of Mathematics, University of Amsterdam, P.O. Box 94248, Amsterdam, 1090 GE, The Netherlands. Email: B.Kleijn@uva.nl
**Postal address: Department of Statistics, University of Padova, Via Battisti, 241, Padova, 35121, Italy. Email: stefano.rizzelli@unipd.it
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Abstract

Asymptotic properties of random graph sequences, like the occurrence of a giant component or full connectivity in Erdös–Rényi graphs, are usually derived with very specific choices for the defining parameters. The question arises as to what extent those parameter choices may be perturbed without losing the asymptotic property. For two sequences of graph distributions, asymptotic equivalence (convergence in total variation) and contiguity have been considered by Janson (2010) and others; here we use so-called remote contiguity to show that connectivity properties are preserved in more heavily perturbed Erdös–Rényi graphs. The techniques we demonstrate here with random graphs also extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.

Keywords

Random graphscontiguityremote contiguitygraph connectivitygiant component

MSC classification

Primary: 05C80: Random graphs 05C40: Connectivity
Secondary: 60B10: Convergence of probability measures 60G30: Continuity and singularity of induced measures

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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