Sufficiently dense Kuramoto networks are globally synchronizing

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: They stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by markedly reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay

Other than the fully synchronized state, a twisted state can also be an equilibrium solution in the Kuramoto model and its variations. In the present work, we explore the stability of the twisted state in Kuramoto oscillators put on a rim of a planar circle in the two-dimensional space in the presence of power-law decaying interaction strength (∼r^{-α} with the distance r) and time delays due to a finite speed of information transfer. For example, our model can phenomenologically mimic a large sports stadium where many people try to sing or clap their hands in unison; the sound intensity decays with the distance and there can exist a time delay proportional to the distance due to the finiteness of sound speed. We first consider the case without the time delay effect and numerically find that stable twisted states emerge when the exponent α exceeds a critical value of α_{c}≈2. In other words, for α<α_{c}, the fully synchronized state, not the twisted state, is the only stable fixed point of the dynamics. In our analytic approach, we also derive an equation for α_{c} and discuss its solutions. In the presence of time delay, we find that it is possible that the synchronized state becomes unstable while twisted states are stable.

Large Coherent States Formed from Disordered k-Regular Random Graphs

The present work is motivated by the need for robust, large-scale coherent states that can play possible roles as quantum resources. A challenge is that large, complex systems tend to be fragile. However, emergent phenomena in classical systems tend to become more robust with scale. Do these classical systems inspire ways to think about robust quantum networks? This question is studied by characterizing the complex quantum states produced by mapping interactions between a set of qubits from structure in graphs. We focus on maps based on k-regular random graphs where many edges were randomly deleted. We ask how many edge deletions can be tolerated. Surprisingly, it was found that the emergent coherent state characteristic of these graphs was robust to a substantial number of edge deletions. The analysis considers the possible role of the expander property of k-regular random graphs.

On the number of stable solutions in the Kuramoto model

We consider a system of n coupled oscillators described by the Kuramoto model with the dynamics given by θ˙=ω+Kf(θ). In this system, an equilibrium solution θ∗ is considered stable when ω+Kf(θ∗)=0, and the Jacobian matrix Df(θ∗) has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of Df(θ∗) are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that |Γ(θ∗)|≤π, where |Γ(θ∗)| represents the length of the shortest arc on the unit circle that contains the equilibrium solution θ∗. We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system.

Learning to predict synchronization of coupled oscillators on randomly generated graphs

Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph structure, this is an important yet analytically intractable question in general. In this work, we take an alternative approach to the synchronization prediction problem by viewing it as a classification problem based on the fact that any given system will eventually synchronize or converge to a non-synchronizing limit cycle. By only using some basic statistics of the underlying graphs such as edge density and diameter, our method can achieve perfect accuracy when there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples. However, in the problem setting where these graph statistics cannot distinguish the two classes very well (e.g., when the graphs are generated from the same random graph model), we find that pairing a few iterations of the initial dynamics along with the graph statistics as the input to our classification algorithms can lead to significant improvement in accuracy; far exceeding what is known by the classical oscillator theory. More surprisingly, we find that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy. We demonstrate our method on three models of continuous and discrete coupled oscillators-the Kuramoto model, Firefly Cellular Automata, and Greenberg-Hastings model. Finally, we also propose an "ensemble prediction" algorithm that successfully scales our method to large graphs by training on dynamics observed from multiple random subgraphs.

A global synchronization theorem for oscillators on a random graph

Consider n identical Kuramoto oscillators on a random graph. Specifically, consider Erdős-Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability 0 ≤ p ≤ 1. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for p above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, p ∼ log ⁡ ( n ) / n for n ≫ 1. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if p ≫ log ⁡ ( n ) / ⁿ, then Erdős-Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as n → ∞. Here, we improve that result by showing that p ≫ ^log ⁡ ( n ) / n suffices. Our estimates are explicit: for example, we can say that there is more than a 99.9996 % chance that a random network with n = ¹⁰ and p > 0.011 17 is globally synchronizing.