The critical Karp-Sipser core of random graphs
Le coeur de Karp-Sipser des graphes aléatoires
Résumé
We study the Karp-Sipser core of a random graph made of a configuration model with vertices of degree 1, 2 and 3. This core is obtained by recursively removing the leaves as well as their unique neighbors in the graph. We settle a conjecture of Bauer & Golinelli [2] and prove that at criticality, the Karp-Sipser core has size ≈ Cst • ϑ −2 • n 3/5 where ϑ is the hitting time of the curve t → 1 t 2 by a linear Brownian motion started at 0. Our proof relies on a detailed multi-scale analysis of the Markov chain associated to the Karp-Sipser leaf-removal algorithm close to its extinction time. Figure 1: (Left). The giant component of an Erdős-Rényi random graph G(n, e n) with n = 2000 on the left and (Middle) its Karp-Sipser core. (Right). The Karp-Sipser core in red inside the original graph.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
Licence : CC BY - Paternité
Licence : CC BY - Paternité