Complete Graph Identification in Population Protocols

TL;DR

This paper investigates the problem of agents in population protocols determining if their communication graph is complete, providing solutions under certain fairness and knowledge assumptions and proving impossibility results in other cases.

Contribution

It introduces algorithms for complete graph identification with specific state complexities under different fairness and knowledge conditions, and establishes impossibility results in some scenarios.

Findings

O(n^2) states suffice under global fairness with no prior knowledge.

O(n) states suffice under weak fairness with exact population size known.

Impossible to solve with only an upper bound on population size under weak fairness.

Abstract

We consider the population protocol model where indistinguishable state machines, referred to as agents, communicate in pairs. The communication graph specifies potential interactions (\ie communication) between agent pairs. This paper addresses the complete graph identification problem, requiring agents to determine if their communication graph is a clique or not. We evaluate various settings based on: (i) the fairness preserved by the adversarial scheduler -- either global fairness or weak fairness, and (ii) the knowledge provided to agents beforehand -- either the exact population size $n$, a common upper bound $P$ on $n$, or no prior information. Positively, we show that $O(n^2)$ states per agent suffice to solve the complete graph identification problem under global fairness without prior knowledge. With prior knowledge of $n$, agents can solve the problem using only $O(n)$ states under weak fairness. Negatively, we prove that complete graph identification remains unsolvable under weak fairness when only a common upper bound $P$ on the population size $n$ is known.