Constrained Information Transmission on Erdos - Renyi Graphs

F. Comets, C. Gallesco, S.Yu. Popov, M. Vachkovskaia

2016, v.22, №1, 111-138

ABSTRACT

We model the transmission of information of a message on the Erd\"os\tire R\'enyi random graph with parameters $(n,p)$ and limited resources.
The vertices of the graph represent servers that may broadcast a message at random. Each server has a random emission capital that decreases by one at each emission. We examine two natural dynamics: in the first dynamics, an informed server performs all its attempts, then checks
at each of them if the corresponding edge is open or not; in the second dynamics the informed server knows a priori
who are its neighbors, and it performs all its attempts on its actual neighbors in the graph.
In each case, we obtain first and second order asymptotics
(law of large numbers and central limit theorem), when $n\to \infty$ and $p$ is fixed, for the final proportion of informed servers.

Keywords: information transmission, rumor, labelled trees, Erd\"os\tire R\'enyi random graph