Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

A linear system $\dot{x}=Ax$, $A\in {\mathbb{R}}^{n\times n}$, $x \in $\mathbb{R}$^n$,with$$\mathrm{r}\mathrm{k}$ A = n-1$, has a one-dimensional center manifold $E^c=\{v\in {\mathbb{R}}^n:Av=0\}$. If a differential equation $\dot{x}=f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^{\ast }$ then $E^c$ is tangential to $W^c$ with $A=Df(x^{\ast })$ and for stability of $W^c$ it is necessary that $A$ has no spectrum in ${\mathbb{C}}^{+}$, i.e.\ if $A$ is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to $A$, we formulate meso-scale conditions with certain principal minors of $A$ which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.