Topology and higher-order global synchronization on directed and hollow simplicial and cell complexes

TL;DR

The paper investigates Global Topological Synchronization (GTS) on generalized higher-order networks built from directed (DSC/DCC) and hollow (HSC/HCC) complexes, using boundary operators and Hodge Laplacians to define topology-driven coupling. It shows that GTS can always exist on directed complexes but is not asymptotically stable due to kernel degeneracy, while hollow constructions can enhance both existence and stability for certain topologies; tessellated hollow representations can, in some cases, prevent GTS. The approach combines a Stuart–Landau model for topological signals with Master Stability Function analysis to derive precise conditions ${L}_{[n]}^{(G)}{\bf u}=0$ and ${\bf B}^{(G)\top}{\bf u}=0$, and it provides illustrative examples on lattice-like manifolds (e.g., torus tessellations). These results deepen understanding of how higher-order topology and directionality shape collective dynamics with potential implications for neuroscience, transport networks, and topology-aware AI.

Abstract

Higher-order networks encode the many-body interactions of complex systems ranging from the brain to biological transportation networks. Simplicial and cell complexes are ideal higher-order network representations for investigating higher-order topological dynamics where dynamical variables are not only associated with nodes, but also with edges, triangles, and higher-order simplices and cells. Global Topological Synchronization (GTS) refers to the dynamical state in which identical oscillators associated with higher-dimensional simplices and cells oscillate in unison. On standard unweighted and undirected complexes this dynamical state can be achieved only under strict topological and combinatorial conditions on the underlying discrete support. In this work we consider generalized higher-order network representations including directed and hollow complexes. Based on an in depth investigation of their topology defined by their associated algebraic topology operators and Betti numbers, we determine under which conditions GTS can be observed. We show that directed complexes always admit a global topological synchronization state independently of their topology and structure. However, we demonstrate that for directed complexes this dynamical state cannot be asymptotically stable. While hollow complexes require more stringent topological conditions to sustain global topological synchronization, these topologies can favor both the existence and the stability of global topological synchronization with respect to undirected and unweighted complexes.

Figures (10)

• Figure 1: Schematic illustration of the directed simplicial complex (DSC), hollow simplicial complex (HSC) and tessellated hollow simplicial complex (TSCC). Panel (a) shows a DSC composed of simplices (nodes, edges, and triangles) having single orientation. Note that for simplicity we illustrate only a triangle with single orientation, but the DSC includes two single oriented triangles with opposite orientation. Panel (b) is a HSC, and panel (c) is a THSC induced from the HSC shown in panel (b). Below each of these generalized complexes we indicate their corresponding Betti numbers following Eqs.(\ref{['betti_D']}), (\ref{['betti_H']}),(\ref{['betti_T']}) respectively.
• Figure 2: Illustration of the topology of the HSC as captured by their associated Betti number $\beta^{(H)}$ given by Eqs.(\ref{['betti_H']}) whose derivation is given in \ref{['ApA2']}. Panel (a) shows an original pure $D=2$ dimensional simplicial complex, panel (b) shows associated HSC. The Betti numbers $\beta_n$ is the original network (panel (a)) and the Betti numbers $\beta_n^{(H)}$ of the HSC (panel (b)) are indicated below the corresponding simplicial complex visualizations.
• Figure 3: Schematic illustration of three types cell complexes which generalize the DSC, HSC and THSC shown in Figure \ref{['fig1']}. Panel (a) shows a directed cell complex (DCC) composed of faces, (nodes, edges and squares) having single orientation. Note that for simplicity we illustrate only a square with single orientation, but the DSC includes two single oriented squares with opposite orientation. Panel (b) is a hollow cell complex (HCC), and panel (c) is a tesselated hollow cell complex (THCC) induced from the HCC shown in panel (b). Below each of these generalized complexes we indicate their corresponding Betti numbers following Eqs.(\ref{['betti_D']}), (\ref{['betti_H']}),(\ref{['betti_T']}) respectively.
• Figure 4: Harmonic eigenstates of the square lattice tessellation of a $2D$ torus and stability of the GTS defined on it. Panels (a)-(c) represent respectively the chosen orientation of the faces of the $2D$ torus (panel (a)) and the two orthogonal eigenvectors of type ${\bf u}$ in the kernel of the Hodge Laplacian ${\bf L}_{[1]}$. Panel (b) represents the eigenvector with elements defined on both $x$-edges and $y$-edges equal to one. Panel (c) represents the eigenvector in which elements defined on the $x$-edges are equal to one, while elements define on $y$-edges are equal to minus one. Panels (d)-(f) demonstrate that GTS can be achieved ($R_2$ converges to $1$) when the random initial condition is unbiased on both $x$ and $y$ links. This is due to the fact that in this case the dynamics, to a very good approximation, includes only perturbations orthogonal to the kernel of ${\bf L}_{[1]}$. Panels (g)-(i) demonstrate that instead when the initial condition on the edge signal is biased ($R_2$ converges to $1$) the neutral stability of GTS becomes apparent and GTS cannot be achieved. However our results reveal that in both cases the topological oscillators associated to $x$-edges and the one associated to $y$-edges do synchronize among themselves as indicated by $R_{xy}$ converging to $1$ in both cases. The phase $\theta_{\alpha}$ associated with the complex valued topological signal $\phi_{\alpha}$ defined on each edge $\alpha$ is plotted on each edge of the torus for the unbiased (panel (d)) and the biased (panel (g)) initial condition. The corresponding distributions of the phases $\theta_{\alpha}$ associated to $x$-type edges and $y$-type edges are shown in panels (e) and (h). The resulting dynamics of the generalized order parameters $R_2$ and $R_{xy}$ are shown as a function of time $t$ in panels (f) and (i).The considered dynamics is given by the SL model with parameters $\delta=40+4.3\textrm{i},\mu=40+40.1\textrm{i},\sigma=1+1.1\textrm{i},m=3$.
• Figure 5: Illustration of the properties of the GTS of edge signals on a directed simplicial complex on which the SL dynamics is considered. Panel (a) displays a visualization of a random DSC. Panel (b) presents the results of the MSF approach for edge topological signal on this DSC by showing the largest real part of the Floquet eigenvalues $\lambda_k$ associated to the $k$-th eigenmode of the Hodge Laplacian ${\bf L}_{[1]}$ with eigenvalue $\Lambda^{(k)}$. These results implies that since $\lambda_k$ for $k\neq 0$ are negative, the GTS is stable with respect to perturbation orthogonal to the kernel of ${\bf L}_{[1]}$. Panel (c) demonstrates that for initial conditions that have a non-negligible projection on a single harmonic eigenvector of type ${\bf u}$, the GTS is asymptotically stable. In this case the generalized order parameter $R_2$ convergences to $1$. However panel (d) demonstrates that for more general initial conditions the GTS is only neutrally stable which $R_2$ not converging to $1$ for sufficiently large times $t$. What happens in both scenarios is instead that the oscillators corresponding to the same undirected simplex, oscillate in pairs as revealed by the directed order parameter $R_D$ converging to $R_D=1$ in both panel (c) and (d). The parameters for the SL model are $\delta=40+4.3\textrm{i},\mu=40+40.1\textrm{i},\sigma=1+1.1\textrm{i},m=3$.