  Studying the landscape of nonconvex cost function is key towards a better
understanding of optimization algorithms widely used in signal processing,
statistics, and machine learning. Meanwhile, the famous Kuramoto model has been
an important mathematical model to study the synchronization phenomena of
coupled oscillators over various network topologies. In this paper, we bring
together these two seemingly unrelated objects by investigating the
optimization landscape of a nonlinear function $E(\boldsymbol{\theta}) =
\frac{1}{2}\sum_{1\leq i,j\leq n} a_{ij}(1-\cos(\theta_i - \theta_j))$
associated to an underlying network and exploring the relationship between the
existence of local minima and network topology. This function arises naturally
in Burer-Monteiro method applied to $\mathbb{Z}_2$ synchronization as well as
matrix completion on the torus. Moreover, it corresponds to the energy function
of the homogeneous Kuramoto model on complex networks for coupled oscillators.
We prove the minimizer of the energy function is unique up to a global
translation under deterministic dense graphs and Erd\H{o}s-R\'enyi random
graphs with tools from optimization and random matrix theory. Consequently, the
stable equilibrium of the corresponding homogeneous Kuramoto model is unique
and the basin of attraction for the synchronous state of these coupled
oscillators is the whole phase space minus a set of measure zero. In addition,
our results address when the Burer-Monteiro method recovers the ground truth
exactly from highly incomplete observations in $\mathbb{Z}_2$ synchronization
and shed light on the robustness of nonconvex optimization algorithms against
certain types of so-called monotone adversaries. Numerical simulations are
performed to illustrate our results.
