  Graph Laplacians computed from weighted adjacency matrices are widely used to
identify geometric structure in data, and clusters in particular; their
spectral properties play a central role in a number of unsupervised and
semi-supervised learning algorithms. When suitably scaled, graph Laplacians
approach limiting continuum operators in the large data limit. Studying these
limiting operators, therefore, sheds light on learning algorithms. This paper
is devoted to the study of a parameterized family of divergence form elliptic
operators that arise as the large data limit of graph Laplacians. The link
between a three-parameter family of graph Laplacians and a three-parameter
family of differential operators is explained. The spectral properties of these
differential operators are analyzed in the situation where the data comprises
two nearly separated clusters, in a sense which is made precise. In particular,
we investigate how the spectral gap depends on the three parameters entering
the graph Laplacian, and on a parameter measuring the size of the perturbation
from the perfectly clustered case. Numerical results are presented which
exemplify and extend the analysis: the computations study situations in which
there are two nearly separated clusters, but which violate the assumptions used
in our theory; situations in which more than two clusters are present, also
going beyond our theory; and situations which demonstrate the relevance of our
studies of differential operators for the understanding of finite data problems
via the graph Laplacian. The findings provide insight into parameter choices
made in learning algorithms which are based on weighted adjacency matrices;
they also provide the basis for analysis of the consistency of various
unsupervised and semi-supervised learning algorithms, in the large data limit.
