  Rank-based approaches are among the most popular nonparametric methods for
univariate data in tackling statistical problems such as hypothesis testing due
to their robustness and effectiveness. However, they are unsatisfactory for
more complex data. In the era of big data, high-dimensional and non-Euclidean
data, such as networks and images, are ubiquitous and pose challenges for
statistical analysis. Existing multivariate ranks such as component-wise,
spatial, and depth-based ranks do not apply to non-Euclidean data and have
limited performance for high-dimensional data. Instead of dealing with the
ranks of observations, we propose two types of ranks applicable to complex data
based on a similarity graph constructed on observations: a graph-induced rank
defined by the inductive nature of the graph and an overall rank defined by the
weight of edges in the graph. To illustrate their utilization, both the new
ranks are used to construct test statistics for the two-sample hypothesis
testing, which converge to the $\chi_2^2$ distribution under the permutation
null distribution and some mild conditions of the ranks, enabling an easy
type-I error control. Simulation studies show that the new method exhibits good
power under a wide range of alternatives compared to existing methods. The new
test is illustrated on the New York City taxi data for comparing travel
patterns in consecutive months and a brain network dataset comparing male and
female subjects.
