We consider point clouds obtained as random samples of a measure on a
Euclidean domain. A graph representing the point cloud is obtained by assigning
weights to edges based on the distance between the points they connect. Our
goal is to develop mathematical tools needed to study the consistency, as the
number of available data points increases, of graph-based machine learning
algorithms for tasks such as clustering. In particular, we study when is the
cut capacity, and more generally total variation, on these graphs a good
approximation of the perimeter (total variation) in the continuum setting. We
address this question in the setting of Γ-convergence. We obtain almost
optimal conditions on the scaling, as number of points increases, of the size
of the neighborhood over which the points are connected by an edge for the
Γ-convergence to hold. Taking the limit is enabled by a transportation
based metric which allows to suitably compare functionals defined on different
point clouds