Topological cluster synchronization via Dirac spectral programming on directed hypergraphs

Collective synchronization in complex systems arises from the interplay between topology and dynamics, yet how to design and control such patterns in higher-order networks remains unclear. Here we show that a Dirac spectral programming framework enables programmable topological cluster synchronization on directed hypergraphs. By encoding tail-head hyperedges into a topological Dirac operator and introducing a tunable mass term, we obtain a spectrum whose isolated eigenvalues correspond to distinct synchronization clusters defined jointly on nodes and hyperedges. Selecting a target eigenvalue allows the system to self-organize toward the associated cluster state without modifying the underlying hypergraph structure. Simulations on directed-hypergraph block models and empirical systems–including higher-order contact networks and the ABIDE functional brain network–confirm that spectral selection alone determines the accessible synchronization patterns. Our results establish a general and interpretable route for controlling collective dynamics in directed higher-order systems.

💡 Research Summary

The paper introduces a novel framework called Dirac‑Equation Synchronization Dynamics (DESD) that enables programmable cluster synchronization on directed hypergraphs. By representing a directed hypergraph through a degree‑balanced boundary matrix (B) (which captures the oriented flow from tail to head node sets) and constructing the topological Dirac operator (D=\begin{pmatrix}0&B\ B^{\top}&0\end{pmatrix}), the authors obtain a Hamiltonian (H = D + m\gamma) where (\gamma=\mathrm{diag}(I_{|V|},-I_{|E|})) and (m) is a tunable mass term. The mass creates a spectral gap that separates bulk modes from isolated eigenvalues (|E|>m). Each isolated eigenvalue corresponds to a Dirac eigenmode that is strongly localized on a mesoscopic structural feature (a cluster) of the hypergraph.

The dynamics of node phases (\theta) and hyperedge phases (\phi) are governed by
\