Designing topological cluster synchronization patterns with the Dirac operator

TL;DR

This work addresses the design of stable cluster synchronization in networks by extending dynamics to topological signals on both nodes and edges via the Topological Dirac operator. It introduces Dirac-Equation Synchronization Dynamics (DESD), a free-energy–driven model where the ground state is an eigenstate of the Topological Dirac Equation with energy $\bar{E}$, guiding node/edge dynamics to synchronize along the corresponding eigenmode. A linear stability analysis links stability to the Dirac spectrum through spectral gaps $\Delta_\pm$ and density-of-states exponents $\delta_\pm$, with numerical demonstrations on SBM, random graphs, and a connectome showing stable DESD aligned to isolated eigenstates and modular structure. The results reveal that DESD can realize multiple topology-consistent topological cluster synchronization patterns on the same network, offering a bridge between anatomical connectivity and functional organization and suggesting extensions to higher-order signals and edge-weight design.

Abstract

Designing stable cluster synchronization patterns is a fundamental challenge in nonlinear dynamics of networks with great relevance to understanding neuronal and brain dynamics. So far, cluster synchronization has been studied exclusively in a node-based dynamical approach, according to which oscillators are associated only with the nodes of the network. Here, we propose a topological synchronization dynamics model based on the use of the Topological Dirac operator, which allows us to design cluster synchronization patterns for topological oscillators associated with both nodes and edges of a network. In particular, by modulating the ground state of the free energy associated with the dynamical model, we construct topological cluster synchronization patterns. These are aligned with the eigenstates of the Topological Dirac Equation that provide a very useful decomposition of the dynamical state of node and edge signals associated with the network. We use linear stability analysis to predict the stability of the topological cluster synchronization patterns and provide numerical evidence of the ability to design several stable topological cluster synchronization states on real connectome data, random graphs, and on stochastic block models.

Figures (10)

• Figure 1: Schematic representation of a network and its dynamical state encoded in the topological spinor $\bm\Psi$ comprising the node signals $\bm\theta\in \mathbb{R}^{N_0}$ and the edge signals $\bm\phi\in \mathbb{R}^{N_1}$. Here the node signal associated with node $i$ (0-simplex) of the network is indicated with $\theta_{[i]}(t)$, while the edge signal associated with each edge $\ell=[i,j]$ (1-simplex) of the network is indicated with $\phi_{\ell}(t)$.
• Figure 2: Illustration of the application of DESD to the brain structural connectome obtained by diffusion-weighted MRI in Ref.vskoch2022human. On this network, DESD allows to design two stable topological cluster synchronization patterns aligned along the first two isolated eigenstate of the Topological Dirac Equation $\bar{E}=1.2392\ldots$ (panels (a)-(d)) and $\bar{E}=1.5053\ldots$ (panels (e)-(h)), of the network. These two topological cluster synchronization patterns partition nodes (panels (a) and (e)) and edges (panels (b) and (f) of the connectome revealing its symmetries and modular structure. In panels (a) and (e), the nodes are colored according to the associated component $\Psi_i^{(\bar{E})}$ of the selected eigenstate of the Topological Dirac Equation revealing in panel (a), the left (red) and right (blue) hemispheres, and in panel (e) the anterior (red) and posterior (blue) parts. In panels (b) and (f), the edges are colored according to the component $\Psi_{\ell}^{(\bar{E})}$ of the selected eigenstate of the Topological Dirac Equation revealing distinct fiber bundles mediating interactions between these functional modules. Panels (c), (d), (g), and (h) show that the node frequencies $\dot{\theta}_{i}$ and edge frequencies $\dot{\phi}_{\ell}$ associated with the DESD are proportional to the components $\Psi_i^{(\bar{E})}$ and $\Psi_\ell^{(\bar{E})}$ of the selected eigenstate of the Topological Dirac Equation. The simulation of DESD is performed for a mass $m=1$ and a coupling constant $\sigma=15$.
• Figure 3: The spectrum of the Topological Dirac Equations is described by the cumulative distribution of eigenstates $\rho_c(E)$ indicating for positive energy values $E$ the number of eigenstates of energy $E^{\prime}$ with $E^{\prime}\geq E$ and for negative energy values $E$ the number of eigenstates of energy $E^{\prime}$ with $E^{\prime}\leq E$. From this plot, two main observations can be made: (i) the non-zero mass $m$ induces a spectral gap; (ii) the spectrum is symmetric with the only exception of the eigenstates of energy $E$ with $|E|=m$. The data shown here is for a random network with $N_{0}=492$ nodes and average degree $c=3$.
• Figure 4: The DESD along the isolated (gapped) eigenstate associated with energy state $\bar{E}$ is characterized by the eigenmode that oscillates freely, while the eigenmodes corresponding to the other energy states freeze asymptotically in time. Panel (a) shows the time series for $\sin(c_{\bar{E}}(t))$ for the eigenmode at energy $\bar{E}=1.6642\ldots$ corresponding to the isolated Fiedler singular value $\lambda$ of the boundary operator of a Poisson network with $N_{0}=1000$ nodes and average degree $c=12$. Panels (b)-(d) show the time series for $\sin(c_{E}(t))$ corresponding to the three eigenmodes associated to the energies $E$ nearest to $\bar{E}$, namely (b) $E=1.8484\ldots$, (c) $E=1.8797\ldots$, and (d) $E=1.8954\ldots$ on the same network. The simulation of DESD is performed for a mass $m=1$ and a coupling constant $\sigma=15$.
• Figure 5: The existence of a topological DESD is demonstrated by plotting in panel (a) the order parameters $R_{\alpha}$ and $R_{\beta}$ , and in panel (b) the free energy density $f$ as functions of the coupling constant $\sigma$. The selected eigenstate is the Fiedler eigenstate, $\bar{E} = 1.6643$, of the same Poisson network studied in Figure \ref{['fig:sinc']} and the mass is $m=1$. The simulations are performed along the forward transition from random initial conditions. The equilibration time is $T_{\text{max}} = 15$ with time steps of $dt = 0.001$. The values of $R_{\alpha}$, $R_{\beta}$, and $f$ were averaged over the final third of $T_{\text{max}}$ for each $\sigma$.