Multiorder Laplacian for synchronization in higher-order networks

The emergence of synchronization in systems of coupled agents is a pivotal phenomenon in physics, biology, computer science, and neuroscience. Traditionally, interaction systems have been described as networks, where links encode information only on the pairwise influences among the nodes. Yet, in many systems, interactions among the units take place in larger groups. Recent work has shown that the presence of higher-order interactions between oscillators can significantly affect the emerging dynamics. However, these early studies have mostly considered interactions up to four oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this end, we introduce a multiorder Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multiorder Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multiorder Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way toward a general treatment of dynamical processes beyond pairwise interactions.

Figure 1

Population of oscillators with higher-order interactions on simplicial complexes. (a) Example of simplicial complex of oscillators: the red node has higher-order interactions of orders up to 3 (4-oscillator) with the other oscillators. (b) Here, the building blocks of the higher-order interactions consist in edges (1-simplices), triangles (2-simplices), and tetrahedra (3-simplices). A $d$-simplex represents a $(d+1)$-oscillator interaction. (c) The simplicial complex (a) can be decomposed into its pure $d$-simplex interactions: the red node has seven 1-simplex interactions (blue), four 2-simplex interactions (orange), and one 3-simplex interaction (green), but no 4- or more simplex interactions. In (a), the faces of the tetrahedra belong to 2-simplices (orange) and to a 3-simplex (green) so that they are depicted in gray.

Figure 2

Higher-order all-to-all: attractive coupling. Higher-order interactions increase the stability of synchronization. Synchronization of $N=100$ oscillators over time with (a) only 1-simplex (2-oscillator) interactions and (b) only 2-simplex (3-oscillator) interactions, obtained by numerical integration. The oscillators with only 2-simplex interactions synchronize faster than those with only 1-simplex ones. This is confirmed by (c): the analytical second Lyapunov exponent ${\lambda }_2^{\left(d\right)}$ of Eq. (21) as a function of network size $N$, at each order $d=1,\cdots ,4$. The Lyapunov exponent is more negative for larger orders $d$ of interaction. (d) Lyapunov exponent of the full system, as a function of $D$, the largest order taken into account. The higher the order of interactions taken into account, the more negative the Lyapunov exponent is, i.e., the most stable the synchronized state is. Parameters are $N=100$ and ${\gamma }_d=1$ for all $d$.

Figure 3

Higher-order all-to-all: (a)–(d) interplay of attractive and repulsive coupling orders, and (e) decaying coupling strength. Phases over time with (a) which synchronize even with repulsive 1-simplex interactions due to attractive 2-simplex interactions [${\gamma }_1=-0.5$, ${\gamma }_2+-0.5$, black plus in (c)], but do not synchronize with attractive 1-simplex interactions because of repulsive 2-simplex interactions [${\gamma }_1=+0.5$, ${\gamma }_2=-0.5$, black star in (c)]. Time series are numerically integrated on a simplicial complex of $N=100$ nodes. (c) Analytical, nonzero, multiorder Lyapunov exponent (with $D=2$) for a range of 1- and 2-simplex coupling strengths $({\gamma }_1,{\gamma }_2)$. Positive and negative coupling strengths correspond to attractive and repulsive coupling, respectively. Negative values (blue) and positive values (red) of ${\lambda }_2^{\text{(mul)}}$ indicate instability and stability of the synchronized state, and confirm numerics in (a) and (b). The Lyapunov exponent vanishes when ${\gamma }_1+2{\gamma }_2=0$ (dashed black). (d) Analytical, nonzero, multiorder Lyapunov exponent as a function of the largest order taken into account, $D$, for attractive and repulsive coupling at even and odd orders, respectively. The synchronized state changes its stability as higher orders are taken into account. (c) Analytical, nonzero, multiorder Lyapunov exponent as a function of $D$ for decaying coupling strengths ${\gamma }_d={\gamma }_1^d$, with ${\gamma }_1=0.6$. The Lyapunov exponent converges to a negative value as higher-order interactions are taken into account.

Figure 4

Synchronization in the star-clique simplicial complex (a) with $N_s=6$ and $N_c=7$. The highest-order of interaction is $D=6$. In the clique, faces are part of simplices of five different orders, and hence they are depicted in gray. (b) Lyapunov spectra for each case of pure $d$-simplex interactions. (c) Lyapunov spectrum with higher-order interactions combined (black), ${\gamma }_d=1$ for all $d$, compared to the traditional pure 1-simplex case (blue).

Figure 5

Real macaque brain dataset: structure and dynamics. Panels (a), (b), and (c) depict the simplicial complex as it appears at orders $d=1$, 4, and 9, respectively, and they report the associated distributions of degree $K_i^{\left(d\right)}$ across nodes. At each order, most nodes have zero or few interactions, but a few nodes have many. (d) Number of $d$-simplices in the simplicial complex for each $d$. The highest order of interaction in the network is $D=10$. (e) Lyapunov spectra with only 1-simplex interactions, and with higher-order interactions (with $d$ up to $D=4$). (f) Comparison of the Lyapunov spectra: multiorder (black) and projected graph (blue), i.e., $D=1$. The spectrum is affected by higher-order interactions.