Global Topological Dirac Synchronization

TL;DR

This work proposes and investigates Global Topological Dirac Synchronization (GTDS) on higher-order networks such as cell and simplicial complexes and points out that GTDS is a possible dynamical state of cell complexes and simplicial complexes that occur only in some specific network topologies and geometries.

Abstract

Synchronization is a fundamental dynamical state of interacting oscillators, observed in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics as received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where Global Topological Dirac Synchronization can be observed. Our results point out that Global Topological Dirac Synchronization is a possible dynamical state of simplicial and cell complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks

Figures (9)

• Figure 1: The dynamical state of a simplicial complex is encoded in the topological spinor $\mathbf{X}$ given by the direct sum of the topological signals of different dimensions. Thus, the dynamical state of a simplicial complex of dimension $K=2$ (shown in the Figure) is encoded in the topological spinor $\mathcal{X}=(\mathbf{x^{(0)}},\mathbf{x^{(1)}},\mathbf{x^{(2)}})^{\top}$, where $\mathbf{x^{(0)}},$$\mathbf{x^{(1)}},$ and $\mathbf{x^{(2)}}$ indicate the node signals, the edge signals and the triangle signals, respectively, of the simplicial complex.
• Figure 1: Panel (a): The weighted Square Lattice Tessellation of the $2$-Torus (SLTT). We report one basic cell with the oriented edges. The orientation of the square is shown by using the curved arrow. The unweighted square tessellated $2$-torus is obtained by putting $w_1=w_2=1$. Panel (b): The Weighted Triangulated $2$-Torus (WTT). We report one basic cell with the oriented weighted edges. The orientation of each triangle is shown by using the curved arrow.
• Figure 2: Top panels: we report the Master Stability Function, i.e., the maximum of the real part of the characteristic roots, as a function of $b_\alpha$, the singular values of the matrix $\mathbf{B}_1$ ($C_1<0$ left column and $C_1>0$ right column, in both cases $\mu^{(0)}_\Im \mu^{(1)}_\Im>0$). Bottom panels show the order parameter for node and edge signals together with the real part of the node and edge signal, defined on a triangulated $2$-torus. In the left column panels we used the parameters $\mu^{(0)}_\Im = -0.5$, $\mu^{(1)}_\Re = -0.5$ and $\mu^{(1)}_\Im = -0.24$, while in the right column panels $\mu^{(0)}_\Im = -1.5$, $\mu^{(1)}_\Re = -0.75$$\mu^{(1)}_\Im = -1.0$. The remaining parameters are: $\sigma_\Re = 0.2$, $\sigma_\Im = 0.3$, $\beta_\Re = 1.0$, $\beta_\Im = 1.1$, $\mu^{(0)}_\Re = 1.0$.
• Figure 3: We present the sign of the quantities $C_1$ and $\mu^{(0)}_\Im \mu^{(1)}_\Im$ as a function of $\mu^{(0)}_\Im$ and $\mu^{(1)}_\Im$. The remaining parameters have been fixed to $\sigma_\Re = 0.2$, $\sigma_\Im = 0.3$, $\beta_\Re = 1.0$, $\beta_\Im = 1.1$, $\mu^{(0)}_\Re = 1.0$, $\mu^{(1)}_\Re = -0.5$, namely the same used in the left columns of Fig. \ref{['fig:numres']}. The red region corresponds to $C_1<0$ and $\mu^{(0)}_\Im \mu^{(1)}_\Im<0$, the white one to $C_1>0$ and $\mu^{(0)}_\Im \mu^{(1)}_\Im<0$, the blue one to $C_1<0$ and $\mu^{(0)}_\Im \mu^{(1)}_\Im>0$ and the black one to $C_1>0$ and $\mu^{(0)}_\Im \mu^{(1)}_\Im>0$. Let us observe that parameters associated to the blue region allow for global topological synchronization provided $\max b_\alpha$ is small enough.
• Figure 4: We present an extended analysis of the results reported in Fig. \ref{['fig:numresC1C2']}, where the size of the interval of stability, $[0,\hat{b}]$, is depicted by using a color code. Red areas indicate values of $\mu_\Im^{(0,1)}$ that correspond to an unstable region near $0$, namely a positive dispersion relation. Blue values are associated to small $\hat{b}$, while yellow ones to large $\hat{b}$. Those results have been numerically obtained by using a grid search (data and more details are available in the paper repository github_repo).

Theorems & Definitions (2)

• Proposition 1
• Proposition 2