Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes

The higher-order interactions of complex systems, such as the brain, are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Figure 1

An example of a small simplicial complex with the orientation of the simplices induced by the labeling of the nodes.

Figure 2

The projection of the higher-order ($n=1$) Kuramoto dynamics on ($n-1$)-dimensional faces and ($n+1$)-dimensional faces is investigated by plotting the order parameters $R^{[+]}$ (left panel) and $R^{[-]}$ (right panel), both for the simple (blue circles) and explosive (red squares) dynamics. Here both the simple and the explosive higher-order Kuramoto model have $\mathrm{\Omega }=2$ and are defined on a configuration model of $N_{[0]}=1000$ nodes, $N_{[1]}=5299$ links, and $N_{[2]}=4147$ triangles with generalized degree of the nodes that is power-law distributed with power-law exponent $\gamma =2.8$.

Figure 3

The order parameters $R$, $R^{[1]}$, and $R^{[2]}$ of the simple (blue circles) and explosive (red squares) higher-order ($n=1$) Kuramoto dynamics are plotted versus the coupling constant $\sigma$. The network parameters are the same as in Fig. 2.