# Scalability and Fragility in Bounded-Degree Consensus Networks

## TL;DR

This paper examines how the performance and stability of linear consensus algorithms in bounded-degree networks are affected by network size, highlighting scalability issues and fragility in leader-follower configurations, especially in expander graphs.

## Contribution

It provides a detailed analysis of the impact of network size and structure on consensus performance, revealing fundamental limitations and fragility in scalable leader-follower networks.

## Key findings

- Consensus performance deteriorates with network size in bounded-degree graphs.
- Expander graphs maintain good performance but are fragile to grounding.
- Grounding can cause instability or performance loss in large high-order networks.

## Abstract

We investigate the performance of linear consensus algorithms subject to a scaling of the underlying network size. Specifically, we model networked systems with $n^{\text{th}}$ order integrator dynamics over families of undirected, weighted graphs with bounded nodal degrees. In such networks, the algebraic connectivity affects convergence rates, sensitivity, and, for high-order consensus ($n \ge 3$), stability properties. This connectivity scales unfavorably in network size, except in expander families, where consensus performs well regardless of network size. We show, however, that consensus over expander families is fragile to a grounding of the network (resulting in leader-follower consensus). We show that grounding may deteriorate system performance by orders of magnitude in large networks, or cause instability in high-order consensus. Our results, which we illustrate through simulations, also point to a fundamental limitation to the scalability of consensus networks with leaders, which does not apply to leaderless networks.