Connectivity Preserving Network Transformers

Abstract

The Population Protocol model is a distributed model that concerns systems of very weak computational entities that cannot control the way they interact. The model of Network Constructors is a variant of Population Protocols capable of (algorithmically) constructing abstract networks. Both models are characterized by a fundamental inability to terminate. In this work, we investigate the minimal strengthenings of the latter model that could overcome this inability. Our main conclusion is that initial connectivity of the communication topology combined with the ability of the protocol to transform the communication topology and the ability of a node to detect when its degree is equal to a small constant, plus either a unique leader or the ability of detecting common neighbors, are sufficient to guarantee not only termination but also the maximum computational power that one can hope for in this family of models. In particular, the model, under these minimal assumptions, computes with termination any symmetric predicate computable by a Turing Machine of space \(\varTheta (n^2)\).