  The Kuramoto model is fundamental to the study of synchronization. It
consists of a collection of oscillators with interactions given by a network,
which we identify respectively with vertices and edges of a graph. In this
paper, we show that a graph with sufficient expansion must be globally
synchronizing, meaning that a homogeneous Kuramoto model of identical
oscillators on such a graph will converge to the fully synchronized state with
all the oscillators having the same phase, for every initial state up to a set
of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p
\geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the
Erd\H{o}s-R\'enyi random graph $G(n, p)$ is globally synchronizing with
probability tending to one as $n$ goes to infinity. This improves on a previous
result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu,
and Bandeira. We also show that the model is globally synchronizing on any
$d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large
enough degree $d$.
