  The continuum limit provides a useful tool for analyzing coupled oscillator
networks. Recently, Medvedev (Comm. Math. Sci., 17 (2019), no. 4, pp. 883-898)
gave a mathematical foundation for such an approach when the networks are
defined on single graphs which may be dense or sparse, directed or undirected,
and deterministic or random. In this paper, we consider coupled oscillator
networks depending on multiple graphs, and extend his results to show that the
continuum limit is also valid in this situation. Specifically, we prove that
the initial value problem (IVP) of the corresponding continuum limit has a
unique solution under general conditions and that the solution becomes the
limit of those to the IVP of the networks in some adequate meaning. Moreover,
we show that if solutions to the networks are stable or asymptotically stable
when the node number is sufficiently large, then so are the corresponding
solutions to the continuum limit, and that if solutions to the continuum limit
are asymptotically stable, then so are the corresponding solutions to the
networks in some weak meaning as the node number tends to infinity. These
results can also be applied to coupled oscillator networks with multiple
frequencies by regarding the frequencies as a weight matrix of another graph.
We illustrate the theory for three variants of the Kuramoto model along with
numerical simulations.
