Global Topological Dirac Synchronization

TL;DR

This paper introduces the concept of Global Topological Dirac Synchronization in higher-order networks, demonstrating its existence and stability through algebraic topology, non-linear dynamics, and machine learning, with potential implications for complex systems.

Contribution

It proposes a novel synchronization state in higher-order networks using the Topological Dirac operator, combining topology and dynamics to identify conditions for its existence and stability.

Findings

Global Topological Dirac Synchronization observed in 1D and 2D complexes.

Synchronization depends on specific network topologies and geometries.

The study integrates algebraic topology with non-linear dynamics and machine learning.

Abstract

Synchronization is a fundamental dynamical state of interacting oscillators, observed in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics as received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where Global Topological Dirac Synchronization can be observed. Our results point out that Global Topological Dirac Synchronization is a possible dynamical state of simplicial and cell complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks