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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.

"Domaine(s)" et indice(s) Dewey

• Probabilités, Statistiques mathématiques, Mathématiques appliquées (519)

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