Information Transmission under Random Emission Constraints

TL;DR

This paper models message transmission on a complete graph with limited resources, analyzing the number of informed servers and exhaustion time using probabilistic limit theorems as the network size grows.

Contribution

It introduces a novel probabilistic framework connecting message transmission with Galton-Watson trees and coupon collector dynamics, providing rigorous limit theorems.

Findings

Law of large numbers for informed servers

Central limit theorem for exhaustion time

Large deviation principles for transmission process

Abstract

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n tends to infinity, for the proportion of visited vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton-Watson tree with respect to the success epochs of the coupon collector problem.