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

A linear system x' = Ax, with A in n×n, x in R, has a one-dimensional center manifold Ec = {v in Rn : Av = 0}. If a differential equation x' = f(x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to Wc with A = Df(x?) and for stability of Wc it is necessary that A has no spectrum in 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 mesoscale 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.