Hypothesis testing for two sample comparison of non Euclidean network data

Networks are a major type of object data containing numeric and geometrical information that needs to be appropriately parameterized in a non Euclidean space. The development of statistical methodologies for network data is challenging and currently under active investigation. In particular, the non Euclidean counterpart of the two-sample test for network data is still scarce in literature. This study presents a novel framework, NEPTUNE, for comparing two independent samples of non Euclidean networks. Specifically, we elucidate that network data geometry can be properly accounted for in quotient Euclidean space and propose a computationally efficient approximation to the true geodesic by considering a feasible path in such space. A novel distance metric is proposed by combining the geodesic approximation with network spectral distance to quantify both local and global dissimilarity of networks. The permutation test is then adapted to verify the distributional equality of two independent groups of networks. In addition, we conduct a comprehensive study of asymptotic statistical properties. Those results may be useful for other studies that rely on distance-based inference. Comprehensive simulation studies and real applications are conducted to demonstrate the superior performance of our method over selected alternatives. A high-dimensional extension of our method is explored as well.