Two-sample Tests for Random Graphs

The study of networks leads to a wide range of high dimensional inference problems. In most practical scenarios, one needs to draw inference from a small population of large networks. This paper studies hypothesis testing of graphs in the above regime. We consider the problem of testing between two populations of inhomogeneous random graphs defined on the same set of vertices. We propose two test statistics based on comparison of the adjacency matrices of the graphs. We show that the statistics lead to uniformly consistent tests in both the "large graph, small sample" and "small graph, large sample" regimes. We further derive lower bounds on the minimax separation rate for the associated testing problems, and show that the constructed tests are near optimal.