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Poisson approximation for cycles in the generalised random graph

Part of: Graph theory Limit theorems Stochastic processes

Published online by Cambridge University Press:  03 November 2025

Matthias Lienau Open the ORCID record for Matthias Lienau [Opens in a new window]
Matthias Lienau*
Affiliation:
Hamburg University of Technology
*
*Postal address: Am Schwarzenberg-Campus 3, Building E, 21073 Hamburg, Germany. Email: matthias.lienau@tuhh.de
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Abstract

The generalised random graph contains n vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments, and study the point process $\mathcal{C}_n$ on $\{3,4,\ldots\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set A, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets A at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and longest cycles when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung–Lu model and the Norros–Reittu model.

Keywords

Generalised random graphcycle countsPoisson approximationlongest cycleshortest cycle

MSC classification

Primary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Secondary: 60G70: Extreme value theory; extremal processes

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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