Processos de ramificação e aplicações em modelos de transmissão de informação

Abstract

In this work, we study the information transmission models in infinite graphs introduced in \cite{Thecone} and \cite{article}, that is, models of transmission of information on infinite graphs subject to the following rules: (1) at time zero, only the root of the graph has the information, (2) in a time greater than or equal to one, a new vertex is informed and transmits the information to neighbors that are within a finite random neighborhood, and (3) informed vertices remain forever informed. They are considered variants of this process in the spherically symmetrical tree that includes as particular cases the periodic tree and the homogeneous tree. In addition, the model is considered in random trees. In this model, we study phase transition, probability of survival, among other important numerical characteristics for this process. It is also considered the particular case in which the influence radius has a Bernoulli distribution. The proofs are based on comparisons with branching processes.