  Networks of coupled nonlinear oscillators have been used to model circadian
rhythms, flashing fireflies, Josephson junction arrays, high-voltage electric
grids, and many other kinds of self-organizing systems. Recently, several
authors have sought to understand how coupled oscillators behave when they
interact according to a random graph. Here we consider interaction networks
generated by a graphon model known as a $W$-random network, and examine the
dynamics of an infinite number of identical phase oscillators, following an
approach pioneered by Medvedev. We show that with sufficient regularity on $W$,
the solution to the dynamical system over a $W$-random network of size $n$
converges in the $L^{\infty}$ norm to the solution of the continuous graphon
system, with high probability as $n\rightarrow\infty$. This result suggests a
framework for studying synchronization properties in large but finite random
networks. In this paper, we leverage our convergence result in the $L^{\infty}$
norm to prove synchronization results for two classes of identical phase
oscillators on Erd\H{o}s-R\'enyi random graphs. First, we show that the
Kuramoto model on the Erd\H{o}s-R\'enyi graph $G(n, \alpha_n)$ achieves phase
synchronization with high probability as $n$ goes to infinity, if the edge
probability $\alpha_n$ exceeds $(\log n)/n$, the connectivity threshold of an
Erd\H{o}s-R\'enyi random graph. Then we show that the Sakaguchi-Kuramoto model
on the Erd\H{o}s-R\'enyi graph $G(n, p)$ achieves frequency synchronization
with high probability as $n$ goes to infinity, assuming a fixed edge
probability $p\in(0,1]$ and a certain regime for the model's phase shift
parameter. A notable feature of the latter result is that it holds for an
oscillator model whose dynamics are not simply given by a gradient flow.
