Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices

TL;DR

This paper introduces a framework for analyzing the stability of synchronous states in hypernetworks by reducing the problem's dimensionality through simultaneous block-diagonalization of matrices, enabling efficient stability analysis of large networks.

Contribution

The authors develop a novel method for stability analysis of hypernetworks using matrix block-diagonalization, significantly reducing computational complexity for large networks.

Findings

Dimensionality reduction enables stability analysis of large hypernetworks.

Arbitrarily large networks can be reduced to low-dimensional subsystems.

The method is validated through numerical simulations and applied to hypermotifs.

Abstract

We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block-diagonalization of matrices. We obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower-dimensional problems and test the predictions of our low-dimensional analysis through numerical simulations. Under certain conditions, this technique may yield a substantial reduction of the dimensionality of the problem. For example, for a class of dynamical hypernetworks analyzed in the paper, we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2. We apply our reduction techique to a number of different examples, including a class of undirected unweighted hypermotifs of three nodes.