  Scalings in which the graph Laplacian approaches a differential operator in
the large graph limit are used to develop understanding of a number of
algorithms for semi-supervised learning; in particular the extension, to this
graph setting, of the probit algorithm, level set and kriging methods, are
studied. Both optimization and Bayesian approaches are considered, based around
a regularizing quadratic form found from an affine transformation of the
Laplacian, raised to a, possibly fractional, exponent. Conditions on the
parameters defining this quadratic form are identified under which well-defined
limiting continuum analogues of the optimization and Bayesian semi-supervised
learning problems may be found, thereby shedding light on the design of
algorithms in the large graph setting. The large graph limits of the
optimization formulations are tackled through $\Gamma-$convergence, using the
recently introduced $TL^p$ metric. The small labelling noise limits of the
Bayesian formulations are also identified, and contrasted with pre-existing
harmonic function approaches to the problem.
