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<title><string language="fre"><![CDATA[Eternal family trees and dynamics on unimodular random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)]]></string></title>
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<string language="fre"><![CDATA[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.]]></string></description>
<keyword><string language="fre"><![CDATA[processus ponctuels]]></string></keyword><keyword><string language="fre"><![CDATA[graphes aléatoires]]></string></keyword><keyword><string language="fre"><![CDATA[dynamique des réseaux stochastiques]]></string></keyword><keyword><string language="fre"><![CDATA[modélisation réseau]]></string></keyword>
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<entity><![CDATA[BEGIN:VCARD
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