Topological cluster synchronization via Dirac spectral programming on directed hypergraphs

TL;DR

The paper addresses programmable control of collective dynamics in directed higher-order networks by extending the Dirac-Equation Synchronization Dynamics (DESD) to directed hypergraphs. It builds a massive Dirac operator $H=D+m\,\gamma$ from a degree-balanced boundary matrix $B$, enabling spectral targeting of isolated eigenvalues $\bar{E}$ to steer dynamics toward prescribed cluster patterns without altering the underlying topology, with non-target modes suppressed by spectral separation. Through synthetic Hypergraph Stochastic Block Models and two empirical systems (social contact and ABIDE brain networks), the study demonstrates that spectral targeting alone determines accessible synchronization patterns, and introduces a spectral-control mechanism to mitigate mode crowding. The work provides a general, interpretable route to engineer higher-order synchronization in directed hypergraphs, with potential applications in neuroscience, social systems, and beyond.

Abstract

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.

Figures (7)

• Figure 1: From directed hypergraph structure to programmable cluster synchronization.(a) An input directed hypergraph is defined, composed of nodes (circles) and hyperedges (diamonds). (b) The same hypergraph is shown in its final state, with nodes colored according to their emergent dynamical clusters, representing the output of the simulation. (c) The hypergraph's topology and orientation are encoded in a hypergraph boundary matrix, $B$, which encodes the spectral property of the hypergraph. (d) The cumulative density of eigenvalues $\rho_c(E)$, of the system's Dirac operator is calculated. A specific, isolated eigen-energy $\bar{E}$ is selected as the control target. (e) The time evolution of the node frequencies, $\dot{\theta}_{\text{nodes}}$, under the DESD equation. The simulation demonstrates the system's convergence from a disordered state to a stable configuration, where the nodes are clearly partitioned into two synchronized clusters. (f) The theoretical cluster classification derived from the target eigenstate, $\theta^{(\bar{E})}$. The structure of this eigenvector serves as a precise blueprint for the dynamics: its components are directly proportional to the final nodal frequencies achieved in the simulation (e), and its partitioning of the nodes (orange and blue) perfectly matches the final clustered state shown in (b).
• Figure 2: An illustrative directed hypergraph and its Dirac spectrum.a Schematic of the hypergraph, generated from a stochastic model with a pronounced cluster structure. The hypergraph consists of four clusters with 20 nodes each (for a total of $V=80$ nodes). All hyperedges are uniform with a cardinality of $k=3$. The connectivity is governed by the expected intra-cluster degree ($c_{\text{in}}=50$) and inter-cluster degree ($c_{\text{out}}=1$), resulting in dense clusters with sparse connections between them. Nodes are represented by differently coloured circles and the oriented hyperedges by differently coloured rhombuses. b The cumulative density of states, $\rho_c(E)$, of the hypergraph shown in panela is calculated for the massive Dirac operator, $H=D+m\gamma$, with $m=1$. This cumulative density of state $\rho_c(E)$ for positive values of the energy $E$ is counting the fraction of eigenvalues with higher energy, while for negative values of the energy $E$ is counting the fraction of eigenvalues with smaller energy. The spectrum and features a prominent energy gap around $E=0$, which separates the positive and negative energy bands. A isolated eigenstates at ($E \approx\pm 1.845$ ) distinct from the bulk states are clearly noticeable.
• Figure 3: Cluster classification of a hypergraph using isolated eigenstates. This figure demonstrates cluster classification based on three distinct isolated eigenstates of the Hamiltonian constructed from the hypergraph structure shown in Fig. \ref{['Fig.2']} (a). The top row (a--c) shows the classification for nodes, while the bottom row (d--f) shows the classification for hyperedges. Each column corresponds to a classification derived from an eigenstate with a specific isolated eigenvalue, with absolute values of: $E_1 = 1.827$ (left), $E_2 = 1.847$ (middle), and $E_3 = 1.856$ (right). The plots show the components of the node eigenstate ($\theta^{(\bar{E})}$) under the positive energy state and the hyperedge eigenstate ($\phi^{(\bar{E})}$) under the negative energy state, plotted versus the natural frequencies ($\omega$). The colors indicate the cluster assignment for nodes and hyperedges as determined by the structure of each eigenstate.
• Figure 4: Simulated and theoretical cluster synchronization on a hypergraph with heterogeneous cardinality. The top panels (a–c) show node dynamics, and the bottom panels (d–f) show hyperedge dynamics. The hypergraph is characterized by heterogeneous cardinality, with intra-cluster $k_{\text{in}}=3$ ,inter-cluster $k_{\text{out}}=70$,and uniform expected degree $c_{\text{in}}=c_{\text{out}}=25$. The DESD simulation results ,with coupling $\sigma=50$, mass $m=1.0$ and $\alpha=0.1$, which is run up to a max time $T_{max}=5s$, are presented in (a, b, d, e), while the theoretical predictions are in (c, f). a, d Time evolution of the simulated instantaneous frequencies, which rapidly converge. b, e Zoomed-in view of the simulated steady state ($t \in [4.0, 5.0]$), revealing distinct frequency clusters. c, f Theoretical frequency modes predicted by the eigenstates ($\theta^{(\bar{E})}$, $\phi^{(\bar{E})}$) of the hypergraph Hamiltonian. The plotted values are proportional to the mode frequencies, and their cluster structure shows excellent agreement with the simulation.
• Figure 5: Comparison of simulated and theoretical cluster synchronization on a hypergraph. The top panels (a–c) correspond to node dynamics, and the bottom panels (d–f) to hyperedge dynamics. The hypergraph is generated with parameters $c_{\text{in}}=37$, $c_{\text{out}}=1.2$, and uniform cardinality $k=4$. The DESD simulation results, with coupling $\sigma=140$, mass $m=1.0$, and $\alpha=1.0$, which is run up to a max time $T_{max}=30s$, are shown in (a, b, d, e), while the theoretical prediction is shown in (c, f). a, d Time evolution of instantaneous frequencies from the simulation, showing convergence to a steady state. b, e A zoomed-in view of the simulated steady-state frequencies over the interval $t \in [29.0, 30.0]$, revealing distinct frequency clusters. c, f Theoretical prediction for the frequency modes, derived from the eigenstates ($\theta^{(\bar{E})}$ and $\phi^{(\bar{E})}$) of the hypergraph Hamiltonian operator. The plotted values are proportional to the predicted mode frequencies. The resulting cluster structure shows excellent agreement with the simulated frequency clusters in (b, e).