Probabilistic Population Protocol Models

TL;DR

This paper investigates probabilistic population protocols, focusing on confident leader election, and demonstrates that achieving confident leader election is impossible within this model due to the linear interaction requirement.

Contribution

It introduces the probabilistic population protocol model and proves the impossibility of confident leader election within it.

Findings

Linear number of interactions needed for all states to be reachable

Confident leader election is impossible in the probabilistic model

Probabilistic extension does not enable confident leader election

Abstract

Population protocols are a relatively novel computational model in which very resource-limited anonymous agents interact in pairs with the goal of computing predicates. We consider the probabilistic version of this model, which naturally allows to consider the setup in which a small probability of an incorrect output is tolerated. The main focus of this thesis is the question of confident leader election, which is an extension of the regular leader election problem with an extra requirement for the eventual leader to detect its uniqueness. Having a confident leader allows the population protocols to determine the convergence of its computations. This behaviour of the model is highly beneficial and was shown to be feasible when the original model is extended in various ways. We show that it takes a linear in terms of the population size number of interactions for a probabilistic population protocol to have a non-zero fraction of agents in all reachable states, starting from a configuration with all agents in the same state. This leads us to a conclusion that confident leader election is out of reach even with the probabilistic version of the model.