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, has received significant 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 with 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.
Lorenzo Giambagli
PostDoc Department of Physics, Freie Universität Berlin
My research interests include Spectral analysis of Deep Neural Network (DNN), Structura Pruning, Bayesian Inference in DNN, Simplicial Complexes Dynamics, Theoretical Neuroscience