# Random cographs: Brownian graphon limit and asymptotic degree distribution

3 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
7 BioInfo - LRI - Bioinformatique (LRI)
LRI - Laboratoire de Recherche en Informatique
Abstract : We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order $n$, and converges after normalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-02412976
Contributor : Mickaël Maazoun <>
Submitted on : Monday, December 16, 2019 - 9:12:28 AM
Last modification on : Tuesday, December 8, 2020 - 10:42:55 AM

### Identifiers

• HAL Id : hal-02412976, version 1
• ARXIV : 1907.08517

### Citation

Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, Mickaël Maazoun, et al.. Random cographs: Brownian graphon limit and asymptotic degree distribution. 2019. ⟨hal-02412976⟩

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