Multidimensional Asymptotic Consensus in Dynamic Networks

TL;DR

This paper introduces two new algorithms for multidimensional asymptotic consensus in dynamic directed networks, achieving linear convergence time and constant component-wise contraction rates, even under weak connectivity conditions.

Contribution

The paper extends one-dimensional convex combination algorithms to high dimensions with two novel algorithms, the ExtremePoint and Centroid, offering improved convergence properties.

Findings

Both algorithms have constant component-wise contraction rates.

Convergence time is linear in the number of agents.

Algorithms achieve consensus under weak connectivity with bidirectional interactions.

Abstract

We study the problem of asymptotic consensus as it occurs in a wide range of applications in both man-made and natural systems. In particular, we study systems with directed communication graphs that may change over time. We recently proposed a new family of convex combination algorithms in dimension one whose weights depend on the received values and not only on the communication topology. Here, we extend this approach to arbitrarily high dimensions by introducing two new algorithms: the ExtremePoint and the Centroid algorithm. Contrary to classical convex combination algorithms, both have component-wise contraction rates that are constant in the number of agents. Paired with a speed-up technique for convex combination algorithms, we get a convergence time linear in the number of agents, which is optimal. Besides their respective contraction rates, the two algorithms differ in the fact that the Centroid algorithm's update rule is independent of any coordinate system while the ExtremePoint algorithm implicitly assumes a common agreed-upon coordinate system among agents. The latter assumption may be realistic in some man-made multi-agent systems but is highly questionable in systems designed for the modelization of natural phenomena. Finally we prove that our new algorithms also achieve asymptotic consensus under very weak connectivity assumptions, provided that agent interactions are bidirectional.