Synchronization of Dirac-Bianconi driven oscillators

TL;DR

This paper introduces a new framework for analyzing synchronization in higher-order network oscillators driven by Dirac-Bianconi coupling, extending traditional node-based models to include link and higher-dimensional interactions.

Contribution

It develops a phase reduction method for Dirac-Bianconi driven oscillators and demonstrates its application to synchronization analysis in higher-order networks.

Findings

Derived a phase description for Dirac-Bianconi driven oscillators

Analyzed synchronization behavior between two such oscillators

Extended oscillatory analysis beyond node-based network models

Abstract

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac-Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac-Bianconi coupling, hence, which we call it Dirac-Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higher-order networks beyond the node-based paradigm, while providing a ductile modeling tool for node- and edge-signals.