Some qualitative problems in network dynamics

TL;DR

This paper explores qualitative issues in network dynamics, addressing stability and equilibrium characterization in heterogeneous, clustered, and signed networks using graph theory and linear algebra tools.

Contribution

It introduces a flexible analytical framework for stability and equilibrium analysis beyond standard networked dynamical systems, including signed and heterogeneous networks.

Findings

Stability conditions for heterogeneous agent networks.

Characterization of equilibrium sets in signed graphs.

Analysis of non-isolated equilibria with positive and negative weights.

Abstract

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.