Vertex Shifts on Random Graphs

 

Ali Khezeli, Mir-Omid Haji-Mirsadeghi and François Baccelli studied dynamics on the vertices of a random graph in the paper Dynamics on Unimodular Graphs. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils, as in the case of point-shifts on point processes.
The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three types 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).
An infinite connected component of the graph of a vertex-shift on a random network forms an infinite directed tree with one selected end which is referred to as an Eternal Family Tree. An Eternal Family Tree contains a subtree which is a stochastic generalization of a branching process. In a unimodular Eternal Family Tree, the subtree in question is a generalization of a critical branching process. In a $\sigma$-invariant Eternal Family Tree, the subtree is a generalizisation of a non-necessarily critical branching process. The latter trees allow one to analyze dynamics on networks which are not necessarily unimodular.