Synchronization in dynamical networks with unconstrained structure switching

TL;DR

This paper introduces a method to construct and analyze synchronization in dynamical networks with arbitrary, unconstrained topology switching by transforming eigenvector matrices, extending the Master Stability Function formalism.

Contribution

It provides a rigorous approach to model structural evolution in networks with arbitrary switching topologies, independent of their specific structures or time scales.

Findings

Allows stability analysis of synchronized states under arbitrary topology switching

Extends Master Stability Function to non-commuting, fast, or slow topology changes

Provides a framework for smooth eigenvector matrix evolution in network analysis

Abstract

We provide a rigorous solution to the problem of constructing a structural evolution for a network of coupled identical dynamical units that switches between specified topologies without constraints on their structure. The evolution of the structure is determined indirectly, from a carefully built transformation of the eigenvector matrices of the coupling Laplacians, which are guaranteed to change smoothly in time. In turn, this allows to extend the Master Stability Function formalism, which can be used to assess the stability of a synchronized state. This approach is independent from the particular topologies that the network visits, and is not restricted to commuting structures. Also, it does not depend on the time scale of the evolution, which can be faster than, comparable to, or even secular with respect to the the dynamics of the units.