Reviews: Practical Methods for Graph Two-Sample Testing

This paper studies the problem of two-sample testing of large graphs under the inhomogeneous Erdos Renyi model. This model is pretty generic, and assumes that an undirected edge (ij) is in the graph with probability P_{ij} independently of all other edges. Most generically the parameter matrix P could be anything symmetric (zero diagonal), but common models are stochastic block model or mixed membership stochastic block model, which both result in P being low rank. Suppose there were two random graph distributions, parameterized by matrices P and Q, and the goal is to test whether P Q or not (the null hypothesis being that they are equal). They assume that the graphs are vertex-aligned, which helps as it reduces the problem of searching over permutations to align the graphs.