  Ginzburg--Landau (GL) functionals on graphs, which are relaxations of
graph-cut functionals on graphs, have yielded a variety of insights in image
segmentation and graph clustering. In this paper, we study large-graph limits
of GL functionals by taking a functional-analytic view of graphs as nonlocal
kernels. For a graph $W_n$ with $n$ nodes, the corresponding graph GL
functional $\GL^{W_n}_\ep$ is an energy for functions on $W_n$. We minimize GL
functionals on sequences of growing graphs that converge to functions called
graphons. For such sequences of graphs, we show that the graph GL functional
$\Gamma$-converges to a continuous and nonlocal functional that we call the
\emph{graphon GL functional}. We also investigate the sharp-interface limits of
the graph GL and graphon GL functionals, and we relate these limits to a
nonlocal total variation. We express the limiting GL functional in terms of
Young measures and thereby obtain a probabilistic interpretation of the
variational problem in the large-graph limit. Finally, to develop intuition
about the graphon GL functional, we compute the GL minimizer for several
example families of graphons.
