Consensus Networks over Finite Fields

TL;DR

This paper introduces a new class of consensus algorithms over finite fields, enabling finite-time convergence in networks with limited resources, with applications in sensor networks and pose estimation.

Contribution

It defines finite-field consensus dynamics, provides necessary and sufficient conditions for convergence, and proposes a design method and composition rule for such networks.

Findings

Finite-field consensus networks converge in finite time.

Necessary and sufficient conditions depend on network topology and weights.

Application to distributed averaging and pose estimation demonstrated.

Abstract

This work studies consensus strategies for networks of agents with limited memory, computation, and communication capabilities. We assume that agents can process only values from a finite alphabet, and we adopt the framework of finite fields, where the alphabet consists of the integers {0,...,p-1}, for some prime number p, and operations are performed modulo p. Thus, we define a new class of consensus dynamics, which can be exploited in certain applications such as pose estimation in capacity and memory constrained sensor networks. For consensus networks over finite fields, we provide necessary and sufficient conditions on the network topology and weights to ensure convergence. We show that consensus networks over finite fields converge in finite time, a feature that can be hardly achieved over the field of real numbers. For the design of finite-field consensus networks, we propose a general design method, with high computational complexity, and a network composition rule to generate large consensus networks from smaller components. Finally, we discuss the application of finite-field consensus networks to distributed averaging and pose estimation in sensor networks.