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.

我们研究一种无向共识通信协议下的动态网络，并具有一个状态相关的加权边。我们假设上述动态边可以取整个实数的值，并且其行为取决于它所连接的节点和外部慢变量。我们表明，在重量较轻的条件下，存在减少的情况，从而使网络的动力学由跨临界奇点组织。因此，对于一个简单的网络，我们详细介绍了跨临界点的缓慢传递，并且观察到共识和节点群集之间的交换是可能的。与经典的平面快-慢跨临界奇异性相反，所考虑系统的网络结构引起了最大卡纳德现象的出现。我们的主要分析工具是爆炸方法。因此，我们还专注于跟踪爆炸式转换对网络结构的影响。我们显示，在每个爆炸图上，人们都可以恢复与原始人相关的特定动态网络。我们进一步指出了通过跨临界奇点缓慢通过所产生的数值问题。