Stability of synchronization in simplicial complexes

Abstract

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Fig. 1. Synchronization in simplicial complexes of Rössler oscillators.

Contour plots of the time averaged (over an observation time T = 500) synchronization error E (see “Methods” for definition and the vertical bars of each panel for the color code) in the plane (σ₁, σ₂) for some examples of simplicial complexes (whose sketches are reported in the top left of each panel). Simulations refer to coupled Rössler oscillators (x = (x, y, z)^T and f = (−y−z, x+ay, b+z(x−c))^T) with parameters fixed in the chaotic regime (a = b = 0.2, c = 9). In a–d, ${\mathbf{g}}^{\left(1\right)}\left({\mathbf{x}}_i,{\mathbf{x}}_j\right)={\left[x_j-x_i,0,0\right]}^T$, while in (e) ${\mathbf{g}}^{\left(1\right)}\left({\mathbf{x}}_i,{\mathbf{x}}_j\right)={\left[0,y_j-y_i,0\right]}^T$. As for the other coupling function, one has ${\mathbf{g}}^{\left(2\right)}\left({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,y_j^2{y}_k-y_i^3,0\right]}^T$ in (d) and ${\mathbf{g}}^{\left(2\right)}\left({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[x_j^2{x}_k-x_i^3,0,0\right]}^T$ in all other panels. The blue continuous lines are the theoretical predictions of the synchronization thresholds obtained from Eq. (3). a, b, and c are examples of class III problems, whereas panels d and e are examples of class II problems.

Fig. 2. Synchronization in simplicial complexeses of Hindmarsh–Rose neurons.

Contour plots of the time averaged (over an observation time T = 500) synchronization error E (see “Methods” for definition and the vertical bars of each panel for the color code) in the plane (σ₁, σ₂) for simplicial complexes of HR neurons coupled as in Eq. (19). Parameters are fixed in the chaotic regime (r = 0.006, s = 4, I = 3.2). a–c refer to three different simplicial complexes corresponding to the structures considered in Fig. 1a–c. The blue continuous lines are the theoretical predictions of the synchronization thresholds obtained from Eq. (3).

Fig. 3. Synchronization in a simplicial complex of Rössler oscillators with all-to-all coupling.

Lower and upper boundary curves for the region where synchronization is stable, at different values of N. The color codes for the different curves is reported at the top of the panel.

Fig. 4. Synchronization in simplicial complexes of Rössler oscillators, in the case of natural coupling.

The Master Stability Function is here calculated taking into account several coupling functions. a ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[x_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[x_j^2{x}_k,0,0\right]}^T$, b ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[y_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[y_j^2{y}_k,0,0\right]}^T$, c ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[z_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[z_j^2{z}_k,0,0\right]}^T$, d ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,x_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,x_j^2{x}_k,0\right]}^T$, e ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,y_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,y_j^2{y}_k,0\right]}^T$, f ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,z_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,z_j^2{z}_k,0\right]}^T$, g ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,x_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,x_j^2{x}_k\right]}^T$, h ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,y_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,y_j^2{y}_k\right]}^T$, i ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,z_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,z_j^2{z}_k\right]}^T$.

Fig. 5. Synchronization in simplicial complexes of Lorenz systems, in the case of natural coupling.

The Master Stability Function is here calculated taking into account several coupling functions. a ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[x_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[x_j^2{x}_k,0,0\right]}^T$, b ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[y_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[y_j^2{y}_k,0,0\right]}^T$, c ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[z_j^3,0,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[z_j^2{z}_k,0,0\right]}^T$, d ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,x_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,x_j^2{x}_k,0\right]}^T$, e ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,y_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,y_j^2{y}_k,0\right]}^T$, f ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,z_j^3,0\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,z_j^2{z}_k,0\right]}^T$, g ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,x_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,x_j^2{x}_k\right]}^T$, h ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,y_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,y_j^2{y}_k\right]}^T$, i ${\mathbf{h}}^{\left(1\right)}\left({\mathbf{x}}_j\right)={\left[0,0,z_j^3\right]}^T$ and ${\mathbf{h}}^{\left(2\right)}\left({\mathbf{x}}_j,{\mathbf{x}}_k\right)={\left[0,0,z_j^2{z}_k\right]}^T$.

Fig. 6. Synchronization in Zachary karate club structure.

Synchronization is studied in simplicial complexes extracted from the interactions characterizing the Zachary karate club network. a Synchronization error (color code reported in the bar at the right of the panel) vs. σ₁ and σ₂ for the simplicial complex obtained when all the triangles are considered as being 2-simplexes. The red line delimits the area of stability of the synchronous solution predicted by the MSF. b λ₂ vs. the percentage of 2-simplexes in the structure, p_2s (see text for definition); c λ₂/λ_N vs. p_2s. In b and c three different values of r are considered, with the color code for the plotted curves being reported in the corresponding insets.

Fig. 7. Cluster synchronization in simplicial complexes of Rössler oscillators.

a A simplicial complex where the symmetries of ${\mathcal{L}}^{\left(2\right)}$ do not match those of ${\mathcal{L}}^{\left(1\right)}$. b A simplicial complex where the symmetries of ${\mathcal{L}}^{\left(2\right)}$ match those of ${\mathcal{L}}^{\left(1\right)}$. c Synchronization error as a function of σ₁ for the simplicial complex ia. d Synchronization error as a function of σ₁ for the simplicial complex in panel b. In both cases, σ₂ has been set to σ₂ = 0.2.