August 2023 A sample-path large deviation principle for dynamic Erdős–Rényi random graphs
Peter Braunsteins, Frank den Hollander, Michel Mandjes
Author Affiliations +
Ann. Appl. Probab. 33(4): 3278-3320 (August 2023). DOI: 10.1214/22-AAP1892


We consider a dynamic Erdős–Rényi random graph on n vertices in which each edge switches on at rate λ and switches off at rate μ, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as n. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is (n2), the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of d-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.

Funding Statement

The work in this paper was supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.


The authors are grateful to two anonymous referees whose comments helped to improve the paper.


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Peter Braunsteins. Frank den Hollander. Michel Mandjes. "A sample-path large deviation principle for dynamic Erdős–Rényi random graphs." Ann. Appl. Probab. 33 (4) 3278 - 3320, August 2023.


Received: 1 September 2020; Revised: 1 August 2022; Published: August 2023
First available in Project Euclid: 10 July 2023

MathSciNet: MR4612667
zbMATH: 07720505
Digital Object Identifier: 10.1214/22-AAP1892

Primary: 05C80 , 60C05 , 60F10

Keywords: Dynamic random graphs , graphon dynamics , optimal path , Sample-path large deviations

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 4 • August 2023
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