# On fast-slow consensus networks with a dynamic weight

## TL;DR

This paper investigates the dynamics of a consensus network with a single state-dependent weighted edge, revealing a reduction to a transcritical singularity and the emergence of a maximal canard, with implications for consensus and clustering behavior.

## Contribution

It introduces a novel analysis of a dynamic consensus network with a state-dependent weight, demonstrating the occurrence of a transcritical singularity and maximal canard phenomena using the blow-up method.

## Key findings

- Existence of a reduction organized by a transcritical singularity.
- Observation of exchange between consensus and clustering.
- Identification of a numerical issue during slow passage.

## Abstract

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics of the network are organized by a transcritical singularity. As such, we detail a slow passage through a transcritical singularity for a simple network, and we observe that an exchange between consensus and clustering of the nodes is possible. In contrast to the classical planar fast-slow transcritical singularity, the network structure of the system under consideration induces the presence of a maximal canard. Our main tool of analysis is the blow-up method. Thus, we also focus on tracking the effects of the blow-up transformation on the network's structure. We show that on each blow-up chart one recovers a particular dynamic network related to the original one. We further indicate a numerical issue produced by the slow passage through the transcritical singularity.