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- Date de réalisation : 20 Mars 2019
- Lieu de réalisation : Paris
- Durée du programme : 57 min
- Classification Dewey : Probabilités, Statistiques mathématiques, Mathématiques appliquées
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- Catégorie : Conférences
- Niveau : niveau Doctorat (LMD), Recherche
- Disciplines : Mathématiques et informatique, Probabilités
- Collections : ERC Nemo, Workshop Processus ponctuels et graphes aléatoires unimodulaires (20-22 mars 2019)
- ficheLom : Voir la fiche LOM
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- Auteur(s) : Haji-Mirsadeghi Mir-Omid
- producteur : INRIA (Institut national de recherche en informatique et automatique)
- Editeur : INRIA (Institut national de recherche en informatique et automatique) , Baccelli François

Eternal family trees and dynamics on unimodular random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)
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Eternal family trees and dynamics on unimodular random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)
This talk is centered on covariant
dynamics on unimodular random graphs and random networks (marked
graphs), namely maps from the set of vertices to itself which are
preserved by graph or network isomorphisms. Such dynamics are referred
to as vertex-shifts here. These dynamics have point-shifts on point
processes as a subclass. First we give a classification of vertex-shifts
on unimodular random networks. Each such vertex-shift partitions the
vertices into a collection of connected components and foils. The latter
are discrete analogues the stable manifold of the dynamics. The
classification is based on the cardinality of the connected components
and foils. Up to an event of zero probability, there are three classes
of foliations in a connected component: F/F (with finitely many finite
foils), I/F (infinitely many finite foils), and I/I (infinitely many
infinite foils). In the especial case of point-shifts on stationary
point processes the notion of relative intensity can be defined. This
notion formalizes the intuition of invariance of dimension between
consecutive foils and it is the key element to prove this result for the
Hausdorff unimodular dimension of foils. An infinite connected
component of the graph of a vertex-shift on a random network forms an
infinite tree with one selected end which is referred to as an Eternal
Family Tree. Such trees can be seen as stochastic extensions of
branching processes. Unimodular Eternal Family Trees can be seen as
extensions of critical branching processes. The class of
offspring-invariant Eternal Family Trees, allows one to analyze dynamics
on networks which are not necessarily unimodular. These can be seen as
extensions of not necessarily critical branching processes. Several
construction techniques of Eternal Family Trees are proposed, like the
joining of trees or moving the root to a far descendant.
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