  The goal of this paper is to develop a theory of graphon-valued stochastic
processes, and to construct and analyse a natural class of such processes
arising from population genetics. We consider finite populations where
individuals change type according to Wright-Fisher resampling. At any time,
each pair of individuals is linked by an edge with a probability that is given
by a type-connection matrix, whose entries depend on the current empirical type
distribution of the entire population via a fitness function. We show that, in
the large-population-size limit and with an appropriate scaling of time, the
evolution of the associated adjacency matrix converges to a random process in
the space of graphons, driven by the type-connection matrix and the underlying
Wright-Fisher diffusion on the multi-type simplex. In the limit as the number
of types tends to infinity, the limiting process is driven by the
type-connection kernel and the underlying Fleming-Viot diffusion.
