Hypothesis testing for graphs has been an important tool in applied research fields for more than two decades, and still remains a challenging problem as one often needs to draw inference from few replicates of large graphs. Recent studies in statistics and learning theory have provided some theoretical insights about such high-dimensional graph testing problems, but the practicality of the developed theoretical methods remains an open question. In this paper, we consider the problem of two-sample testing of large graphs. We demonstrate the practical merits and limitations of existing theoretical tests and their bootstrapped variants. We also propose two new tests based on asymptotic distributions. We show that these tests are computationally less expensive and, in some cases, more reliable than the existing methods.

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.

Hypothesis testing for graphs has been an important tool in applied research fields for more than two decades, and still remains a challenging problem as one often needs to draw inference from few replicates of large graphs. Recent studies in statistics and learning theory have provided some theoretical insights about such high-dimensional graph testing problems, but the practicality of the developed theoretical methods remains an open question. In this paper, we consider the problem of two-sample testing of large graphs. We demonstrate the practical merits and limitations of existing theoretical tests and their bootstrapped variants. We also propose two new tests based on asymptotic distributions.