Simple and Efficient Local Codes for Distributed Stable Network Construction

TL;DR

This paper introduces minimalistic, finite-state protocols for distributed processes to construct various stable networks through pairwise interactions, providing optimal solutions, lower bounds, and universal protocols capable of simulating Turing Machines.

Contribution

It presents new protocols and bounds for constructing fundamental networks in a minimal, homogeneous, and adversarially scheduled environment, including universal protocols for complex network construction.

Findings

Protocols for spanning line, ring, star, and regular networks with correctness proofs.

Expected convergence times analyzed under uniform random scheduling.

Universal protocols capable of simulating Turing Machines for complex network construction.

Abstract

In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of the their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks.