  Wasserstein gradient flows on probability measures have found a host of
applications in various optimization problems. They typically arise as the
continuum limit of exchangeable particle systems evolving by some mean-field
interaction involving a gradient-type potential. However, in many problems,
such as in multi-layer neural networks, the so-called particles are edge
weights on large graphs whose nodes are exchangeable. Such large graphs are
known to converge to continuum limits called graphons as their size grow to
infinity. We show that the Euclidean gradient flow of a suitable function of
the edge-weights converges to a novel continuum limit given by a curve on the
space of graphons that can be appropriately described as a gradient flow or,
more technically, a curve of maximal slope. Several natural functions on
graphons, such as homomorphism functions and the scalar entropy, are covered by
our set-up, and the examples have been worked out in detail.
