start angle=233, end angle=213\]; ( 3,-4) arc\[radius=5, start angle=307, end angle=327\]; ( 5, 0) circle(0.4); ( 0, 5) circle(0.4); ( 3,-4) circle(0.4); (-3,-4) circle(0.4);

The study of synchronization in networks of coupled oscillators is a particularly pervasive subject in a wide range of independent fields of study in biology, physics, chemistry, engineering, and social science. We refer to [@blekhman_synchronization_1988; @Acebron2005] for a detailed historical account for this topic. The simplest mechanical analog of the coupled oscillator model  is a spring network, shown in Figure \[fig:spring-network\], that consists of a set of weightless particles constrained to move on the unit circle without friction or collision [@dorfler_synchronization_2014]. Here, the coupling strength[^4] $a_{ij} > 0$ characterizes the stiffness of the spring connecting particles $i$ and $j$, and $\frac{d\theta_i}{dt}$ represents the angular velocity (or equivalently, frequency) of the $i$-th particle. Of great interest is the configuration in which the angular velocity of *all* particles can become perfectly aligned, known as *frequency synchronization*. That is, $\frac{d\theta_i}{dt} = c$ for $i=0,\dots,n$ and a constant $c$. Adopting a rotational frame of reference, we can always assume $c = 0$. That is, frequency synchronization configurations are equivalent to equilibria of the Kuramoto model . Under this assumption, the 