Convergence Speed of the Consensus Algorithm with Interference and Sparse Long-Range Connectivity

TL;DR

This paper investigates how interference impacts the convergence speed of consensus algorithms in networks, revealing that increased communication range can either improve or degrade convergence depending on network dimension and interference management.

Contribution

It provides a theoretical analysis of the convergence rate under interference constraints, highlighting the complex trade-offs in network connectivity and interference in multi-dimensional settings.

Findings

In 1D networks, larger communication range speeds up convergence.

In 2D networks, increased range does not affect convergence speed due to offsetting effects.

In higher dimensions, larger range can slow down convergence.

Abstract

We analyze the effect of interference on the convergence rate of average consensus algorithms, which iteratively compute the measurement average by message passing among nodes. It is usually assumed that these algorithms converge faster with a greater exchange of information (i.e., by increased network connectivity) in every iteration. However, when interference is taken into account, it is no longer clear if the rate of convergence increases with network connectivity. We study this problem for randomly-placed consensus-seeking nodes connected through an interference-limited network. We investigate the following questions: (a) How does the rate of convergence vary with increasing communication range of each node? and (b) How does this result change when each node is allowed to communicate with a few selected far-off nodes? When nodes schedule their transmissions to avoid interference, we show that the convergence speed scales with $r^{2-d}$, where $r$ is the communication range and $d$ is the number of dimensions. This scaling is the result of two competing effects when increasing $r$: Increased schedule length for interference-free transmission vs. the speed gain due to improved connectivity. Hence, although one-dimensional networks can converge faster from a greater communication range despite increased interference, the two effects exactly offset one another in two-dimensions. In higher dimensions, increasing the communication range can actually degrade the rate of convergence. Our results thus underline the importance of factoring in the effect of interference in the design of distributed estimation algorithms.