  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.
