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Testing equality of two multivariate distributions is a classical problem forwhich many non-parametric tests have been proposed over the years. Most of thepopular two-sample tests, which are asymptotically distribution-free, are basedeither on geometric graphs constructed using inter-point distances between theobservations (multivariate generalizations of the Wald-Wolfowitz's runs test)or on multivariate data-depth (generalizations of the Mann-Whitney rank test).  This paper introduces a general notion of distribution-free graph-basedtwo-sample tests, and provides a unified framework for analyzing and comparingtheir asymptotic properties. The asymptotic (Pitman) efficiency of a generalgraph-based test is derived, which include tests based on geometric graphs,such as the Friedman-Rafsky test (1979), the test based on the $K$-nearestneighbor graph, the cross-match test (2005), the generalized edge-count test(2017), as well as tests based on multivariate depth functions (the Liu-Singhrank sum statistic (1993)). The results show how the combinatorial propertiesof the underlying graph effect the performance of the associated two-sampletest, and can be used to validate and decide which tests to use in practice.Applications of the results are illustrated both on synthetic and realdatasets.

A General Asymptotic Framework for Distribution-Free Graph-Based Two-Sample Tests