Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators

TL;DR

This paper introduces a novel mathematical approach combining linear algebra, graph theory, and nonlinear dynamics to analyze the complex behavior of multiplex networks of Kuramoto oscillators, enabling decomposition into smaller systems for easier analysis.

Contribution

It presents a new method to decompose multiplex networks into intra- and inter-layer systems, facilitating the study of their dynamics and stability, which was previously an open problem.

Findings

Decomposition of multiplex networks into smaller systems

Construction of solutions from intra- and inter-layer dynamics

Analysis of linear stability of solutions

Abstract

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra and inter-layer system and, in addition, our approach allows us to study the linear stability of these solutions.