Continuum limits of coupled oscillator networks depending on multi sparse graphs

The continuum limits provide useful tools for analyzing coupled oscillator networks.
Recently, Medvedev (Communications in Mathematical Sciences, 17 (2019), no. 4 , 883 -
898) gave a mathematical foundation for such an approach when the networks are defined
on a single graph which may be dense or sparse, directed or undirected, and deterministic
or random. In this paper, we consider coupled oscillator networks depending on multi
graphs, and extend his results to show that the continuum limit is also valid in this
situation. Especially, 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 meanings.
Moreover, we illustrate our theory for some examples.