Steady-State Bifurcations in Coalescence of Motif Networks

Steady-state bifurcation analysis of real-world networks is challenging due to their sizes and complexity. It turns out that, small networks, known as motifs, often serve as fundamental building blocks of larger networks. This raises the question of whether knowledge of steady-state bifurcations in motif networks can provide insights into the steady-state bifurcations in larger networks. In this paper, we explore this question by investigating a particular type of networks obtained by the combination of two motif networks through the coalescence operation, where the two motifs share a common node. We conclude that, in general, bifurcation conditions in a coalescence network are not necessarily bifurcation conditions in its component networks. Consequently, in general, bifurcations in the component networks do not offer clear insights into the bifurcations of the coalescence network. However, for a specific class of networks — feedforward coalescence networks — we prove that the codimension-1 steady-state bifurcations of the coalescence network relate to those in their component networks. In particular, we show that it is possible to infer the bifurcation branches in the coalescence network from those in the component networks. Notably, we conclude that the growth rate of bifurcation branches in the coalescence network depends on the interconnections between the component networks.