  Testing the equality in distributions of multiple samples is a common task in
many fields. However, this problem for high-dimensional or non-Euclidean data
has not been well explored. In this paper, we propose new nonparametric tests
based on a similarity graph constructed on the pooled observations from
multiple samples, and make use of both within-sample edges and between-sample
edges, a straightforward but yet not explored idea. The new tests exhibit
substantial power improvements over existing tests for a wide range of
alternatives. We also study the asymptotic distributions of the test
statistics, offering easy off-the-shelf tools for large datasets. The new tests
are illustrated through an analysis of the age image dataset.
