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2022 Depth first exploration of a configuration model
Nathanaël Enriquez, Gabriel Faraud, Laurent Ménard, Nathan Noiry
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Electron. J. Probab. 27: 1-27 (2022). DOI: 10.1214/22-EJP762

Abstract

We introduce an algorithm that constructs a random graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.

Funding Statement

The first author would like to thank the ANR grants MALIN (Projet- ANR-16-CE93-0003) and PPPP (Projet-ANR-16-CE40-0016) for their financial support. The other three authors would like to thank the ANR grant ProGraM (Projet-ANR-19-CE40-0025) for its financial support. G.F. and L.M. also acknowledge the support of the Labex MME-DII (ANR11-LBX-0023-01).

Acknowledgments

The authors are pleased to thank warmly an anonymous referee for its careful reading, suggestions, and pointing out a mistake in the original manuscript.

Citation

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Nathanaël Enriquez. Gabriel Faraud. Laurent Ménard. Nathan Noiry. "Depth first exploration of a configuration model." Electron. J. Probab. 27 1 - 27, 2022. https://doi.org/10.1214/22-EJP762

Information

Received: 16 December 2020; Accepted: 19 February 2022; Published: 2022
First available in Project Euclid: 27 April 2022

arXiv: 1911.10083
Digital Object Identifier: 10.1214/22-EJP762

Subjects:
Primary: 60F10 , 60J20 , 60K35 , 82C21

Keywords: configuration model , depth first search algorithm , differential equation method

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