Hamiltonian control to desynchronize Kuramoto oscillators with higher-order interactions

TL;DR

This work addresses desynchronization in networks with higher-order interactions by embedding the Higher-Order Kuramoto Model (HOKM) into a Hamiltonian framework and designing an adaptive feedback control via Hamiltonian perturbation theory. The authors construct a Hamiltonian $H(I,\Theta)$ encompassing 2- and 3-body terms and derive control terms $h_i^{(N)}$ and a simplified $\tilde{h}_i^{(N)}$ that act on all or a subset of nodes, respectively. Numerical results show that full control $h^{(N)}$ desynchronizes all-to-all and empirical networks, while pairwise control $\tilde{h}^{(N)}$ suffices in many regimes, particularly when $K_1$ is not too small relative to $K_2$; when $K_1$ is small and $K_2$ large, the higher-order control becomes essential. The study also reveals that the required fraction of controlled nodes to achieve desynchronization depends on topology (roughly $0.6N$ in many cases) and demonstrates the method on all-to-all hypergraphs, random simplicial complexes, and a cat brain connectome, highlighting practical implications for targeted, energy-efficient control of higher-order networks.

Abstract

Synchronization is a ubiquitous phenomenon in nature. Although it is necessary for the functioning of many systems, too much synchronization can also be detrimental, e.g., (partially) synchronized brain patterns support high-level cognitive processes and bodily control, but hypersynchronization can lead to epileptic seizures and tremors, as in neurodegenerative conditions such as Parkinson's disease. Consequently, a critical research question is how to develop effective pinning control methods capable to reduce or modulate synchronization as needed. Although such methods exist to control pairwise-coupled oscillators, there are none for higher-order interactions, despite the increasing evidence of their relevant role in brain dynamics. In this work, we fill this gap by proposing a generalized control method designed to desynchronize Kuramoto oscillators connected through higher-order interactions. Our method embeds a higher-order Kuramoto model into a suitable Hamiltonian flow, and builds up on previous work in Hamiltonian control theory to analytically construct a feedback control mechanism. We numerically show that the proposed method effectively prevents synchronization in synthetics and empirical higher-order networks. Although our findings indicate that pairwise contributions in the feedback loop are often sufficient, the higher-order generalization becomes crucial when pairwise coupling is weak. Finally, we explore the minimum number of controlled nodes required to fully desynchronize oscillators coupled via an all-to-all hypergraphs.

Figures (9)

• Figure 1: Desynchronizing by controlling higher-order interactions in an all-to-all hypergraph. We show the order parameter $R(t)$ over time for (a) $K_1=1$ and (b) $K_1=0.5$, in three cases: uncontrolled (\ref{['eq:HOKM3']}), with only pairwise control $\tilde{h}^{(N)}$ (green), and with full control $h^{(N)}$ (blue). Other parameters are $N=50$ and $K_2=1$. The average order parameter $\hat{R}$ is shown over parameter space in those three cases: (c) no control, (d) full control, and (e) pairwise control. The border of the synchronized region is highlighted by a level curve ($\hat{R}=0.8$) in red. The white cross and plus sign indicate the parameters used in (a) and (b), respectively.
• Figure 2: The control is adaptive. We show (a) the evolution of the order parameter, $R(t)$, in the system with no control (gray) and with full control $h^{(N)}$ (blue), and (b) the intensity of the full control over time. Initially, we set both coupling strengths to small enough values, $K_1=K_2=0.05$: the systems---both uncontrolled and controlled---do not synchronize $R(t)\approx 0$ in the whole time interval, and the intensity of the control is null $I(t)=0$ (b). At $t=15$, we set $K_2=1$ so that the uncontrolled system synchronizes. The control intensity increases and prevents the controlled system from synchronizing. The initial angles are drawn from an uniform distribution, $\theta(0)\sim U([0,2\pi])$.
• Figure 3: Impact of the number of controlled nodes. We report the average Kuramoto order parameter, $\hat{R}$, as a function of the number $M$ of controlled nodes for several values of $K_1$ and $K_2$, with (a) full control $h^{(M)}$, and (b) pairwise control $\tilde{h}^{(M)}$. Each curve represents the average of $150$ independent numerical simulations with different samples for $\pmb{\omega}\sim U([0,1])$ in an all-to-all hypergraph with $N=50$ nodes.
• Figure 4: The case of random simplicial complexes. We show the average order parameter $\hat{R}$ over parameter space $(K_1,K_2)\in[0,10]\times[0,100]$ in three cases: (a) no control, (b) full control $h^{(N)}$, and (c) pairwise control $\tilde{h}^{(N)}$. The random simplicial complex has $50$ nodes and average degrees $\left< k_1\right>=40$ and $\left<k_2\right>=20$. The border of the synchronized region is highlighted by a level curve ($\hat{R}=0.8$) in red.
• Figure 5: Desynchronizing a brain connectome. We show the average order parameter $\hat{R}$ over parameter space $(K_1,K_2)\in[0,5]\times[0,10]$ in three cases: (a) the original model, i.e., without any control, (b) the full control $h^{(N)}$, and (c) the pairwise control $\tilde{h}^{(N)}$. The boundary of the synchronized region is highlighted by the level curve ($\hat{R}=0.8$) in red. (d) Visual representation of the hypergraph. The higher-order brain connectome has $N=65$ nodes, $730$ links and $3613$ triangles.