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This paper addresses analytical aspects of deterministic, continuous-time dynamical systems defined on networks. The goal is to model and analyze certain phenomena which must be framed beyond the context of networked dynamical systems, understood as a set of interdependent dynamical systems defined on the nodes of a (possibly evolving) graph. In order to advocate for a more flexible approach to the study of network dynamics, we tackle some qualitative problems which do not fall in this working scenario. First, we address a stability problem on a network of heterogeneous agents, some of which are of dynamic nature while others just impose restrictions on the system behavior. Our second context assumes that the network is clustered, and we address a two-level stability problem involving the dynamics of both individual agents and groups. The aforementioned problems exhibit lines of non-isolated equilibria, and the analysis implicitly assumes a positiveness condition on the edge weights; the removal of this restriction complicates matters, and our third problem concerns a graph-theoretic characterization of the equilibrium set in the dynamics of certain networks with positive and negative weights, the results applying in particular to signed graphs. Our approach combines graph theory and dynamical systems theory, but also uses specific tools coming from linear algebra, including e.g. Gersgorin discs or Maxwell-type determinantal expansions. The results are of application in social, economic, and flow networks, among others, and are also aimed at motivating further research.

The hyperbolicity problem in circuit theory concerns the existence of purely imaginary eigenvalues (PIEs) in the linearization of the time-domain description of the circuit dynamics. In this paper we characterize the circuit configurations which, in a strictly passive setting, yield purely imaginary eigenvalues for all values of the capacitances and inductances. Our framework is based on branch-oriented, semistate (differential-algebraic) circuit models which capture explicitly the circuit topology, and uses several notions and results from digraph theory. So-called P-structures arising in the analysis turn out to be the key element supporting our results. The analysis is shown to hold not only for classical (RLC) circuits but also for nonlinear circuits including memristors and other mem-devices.