We provide sufficient conditions for the asymptotic normality of the generalized correlation coefficient $sum a_{ij}b_{ij}$ under the permutation null distribution when $a_{ij}$s are symmetric and $b_{ij}$s are symmetric.
Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional data and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse graphs
, such as graphs with the number of edges in the order of the number of observations. However, the tests have better performance with denser graphs under many settings. In this work, we establish the theoretical ground for graph-based tests with graphs that are much denser than those in existing works.
