Constrained information transmission on Erd\"os-R\'enyi graphs

TL;DR

This paper analyzes how information spreads on Erd"os-Rényi graphs with limited resources, deriving asymptotic behaviors for the proportion of informed servers under two different dynamics.

Contribution

It provides the first and second order asymptotics for the final informed proportion in constrained information transmission on Erd"os-Rényi graphs.

Findings

Derived law of large numbers for the final informed proportion.

Established central limit theorem for fluctuations around the mean.

Compared two different transmission dynamics and their asymptotic behaviors.

Abstract

We model the transmission of information of a message on the Erd\"os-R\'eny 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 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.