On the Convergence of Population Protocols When Population Goes to Infinity

TL;DR

This paper investigates how population protocols behave as the population size approaches infinity, revealing that their computational power may differ significantly in this limit, supported by general results and a detailed example.

Contribution

It provides the first mathematical analysis of population protocol convergence as population size tends to infinity, including an asymptotic development for a specific protocol.

Findings

Protocols exhibit different computational capabilities at large population sizes.

General results on convergence are established.

An asymptotic analysis of a particular protocol demonstrates altered computational power.

Abstract

Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement. A population protocol corresponds to a collection of anonymous agents, modeled by finite automata, that interact with one another to carry out computations, by updating their states, using some rules. Their computational power has been investigated under several hypotheses but always when restricted to finite size populations. In particular, predicates stably computable in the original model have been characterized as those definable in Presburger arithmetic. We study mathematically the convergence of population protocols when the size of the population goes to infinity. We do so by giving general results, that we illustrate through the example of a particular population protocol for which we even obtain an asymptotic development. This example shows in particular that these protocols seem to have a rather different computational power when a huge population hypothesis is considered.