Sparse optimization of mutual synchronization in collectively oscillating networks

TL;DR

This paper develops a method to optimize the internetwork coupling in oscillating networks for efficient synchronization, using phase reduction theory and norm minimization to find either dense or sparse coupling structures.

Contribution

It introduces a novel optimization framework for internetwork coupling matrices that enhances synchronization efficiency and resilience, with a focus on sparse coupling via L1-norm minimization.

Findings

Sparse internetwork coupling can be achieved with L1-norm optimization.

Optimized coupling matrices improve synchronization stability.

Different norms lead to either dense or sparse coupling structures.

Abstract

We consider a pair of collectively oscillating networks of dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed in Ref. [H. Nakao, S. Yasui, M. Ota, K. Arai, and Y. Kawamura, Chaos 28, 045103 (2018)]. The dynamical equations describing a pair of weakly coupled networks are reduced to a pair of coupled phase equations, and the linear stability of the synchronized state between the networks is represented as a function of the internetwork coupling matrix. We seek the optimal coupling by minimizing the Frobenius and L1 norms of the internetwork coupling matrix for the prescribed linear stability of the synchronized state. Depending on the norm, either a dense or sparse internetwork coupling yielding efficient mutual synchronization of the networks is obtained. In particular, a sparse yet resilient internetwork coupling is obtained by L1-norm optimization with additional constraints on the individual connection weights.