Structure and stability of the equilibrium set in potential-driven flow networks

In this paper we address local bifurcation properties of a family of networked dynamical systems, specifically those defined by a potential-driven flow on a (directed) graph. These network flows include linear consensus dynamics or Kuramoto models of coupled nonlinear oscillators as particular cases. As it is well-known for consensus systems, these networks exhibit a somehow unconventional dynamical feature, namely, the existence of a line of equilibria, following from a well-known property of the graph Laplacian matrix in connected networks with positive weights. Negative weights, which arise in different contexts (e.g. in consensus models in signed graphs or in Kuramoto models with antagonistic actors), may on the one hand lead to higher-dimensional manifolds of equilibria and, on the other, be responsible for bifurcation phenomena. In this direction, we prove a saddle-node bifurcation theorem for a broad family of potential-driven flows, in networks with one or more negative weights. The goal is to state the conditions in structural terms, that is, in terms of the expressions defining the flowrates and the graph-theoretic properties of the network. Not only the eigenvalue requirements but also the nonlinear transversality assumptions supporting the bifurcation motivate an analysis of independent interest concerning the rank degeneracies of nodal matrices arising in the linearized dynamics; this analysis is performed in terms of the contraction-deletion structure of spanning trees and uses several results from matrix analysis. Different examples illustrate the results; some linear problems (including signed graphs) are aimed at illustrating the analysis of nodal matrices, whereas in a nonlinear framework we apply the characterization of saddle-node bifurcations to networks with a sinusoidal (Kuramoto-like) flow.