# Symmetry Induced Group Consensus

## TL;DR

This paper introduces a novel approach to group consensus in networks where symmetric nodes reach a common state, leveraging network topology to handle heterogeneous agents and achieve consensus on both stable and unstable trajectories.

## Contribution

It presents a new method for symmetry-induced group consensus that exploits network topology, enabling consensus among symmetric nodes in heterogeneous linear agent networks.

## Key findings

- Group consensus can be achieved on both stable and unstable trajectories.
- Symmetric nodes can reach a common final state distinct from others.
- Isolated group consensus phenomena are characterized and observed.

## Abstract

There has been substantial work studying consensus problems for which there is a single common final state, although there are many real-world complex networks for which the complete consensus may be undesirable. More recently, the concept of group consensus whereby subsets of nodes are chosen to reach a common final state distinct from others has been developed, but the methods tend to be independent of the underlying network topology. Here, an alternative type of group consensus is achieved for which nodes that are symmetric achieve a common final state. The dynamic behavior may be distinct between nodes that are not symmetric. We show how group consensus for heterogeneous linear agents can be achieved via a simple coupling protocol that exploits the topology of the network. We see that group consensus is possible on both stable and unstable trajectories. We observe and characterize the phenomenon of isolated group consensus, where one or more clusters may achieve group consensus while the other clusters do not.