  In this paper, we consider the problem of testing the equality of two
multivariate distributions based on geometric graphs constructed using the
interpoint distances between the observations. These include the tests based on
the minimum spanning tree and the $K$-nearest neighbor (NN) graphs, among
others. These tests are asymptotically distribution-free, universally
consistent and computationally efficient, making them particularly useful in
modern applications. However, very little is known about the power properties
of these tests. In this paper, using the theory of stabilizing geometric
graphs, we derive the asymptotic distribution of these tests under general
alternatives, in the Poissonized setting. Using this, the detection threshold
and the limiting local power of the test based on the $K$-NN graph are
obtained, where interesting exponents depending on dimension emerge. This
provides a way to compare and justify the performance of these tests in
different examples.
