Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices

We study how the structure of the coupling matrix $\boldsymbol A$ shapes the energy landscape $E_{\boldsymbol A}(\boldsymbolθ)=-\frac12\sum_{i,j}A_{ij}\cos(θ_i-θ_j)$ of the Kuramoto model. We prove that, for an entrywise nonnegative coupling matrix $\boldsymbol A$ with connected support, if $\boldsymbol A$ has rank-width one over $\mathbb R$, then the landscape is "simple", i.e., every local minimum is synchronized. The rank-width bound is sharp, and the result has many interesting implications for global synchronization. The key contribution of this work is to uncover a structural synchronization mechanism beyond the rank-one regime, which has served as a starting point for earlier analyses of global synchronization.