Topological cluster synchronization on higher-order networks

Investigate topological cluster synchronization on higher-order networks (simplicial complexes) where topological signals are defined not only on nodes and edges but also on higher-dimensional simplices such as triangles and tetrahedra, and ascertain the existence and characteristics of synchronized cluster patterns in this higher-order setting.

Background

The paper introduces Dirac-Equation Synchronization Dynamics (DESD) to design cluster synchronization patterns for topological signals on nodes and edges, leveraging eigenstates of the Topological Dirac Equation. The framework couples node and edge oscillators and predicts stability via linear analysis, demonstrating stable patterns on random graphs and stochastic block models when the targeted eigenstate is spectrally isolated.

Extending these ideas beyond graphs requires handling topological signals on higher-dimensional simplices (e.g., triangles, tetrahedra) typical of simplicial complexes. The authors explicitly flag the need to investigate whether analogous cluster synchronization phenomena arise in these higher-order structures, where new operators and couplings may be needed to capture interactions among higher-dimensional signals.