Two-Sample Testing for Random Graphs without Vertex Correspondence

Two populations of graphs often have to be compared without any correspondence between their vertices, for instance when networks come from different communities, or when a graph generative model is evaluated against held-out graphs. We study how many graphs such an unaligned two-sample test needs, and which graph statistics can detect which differences. For an ErdHos--R'enyi null and a planted two-block difference that leaves every expected degree unchanged, we show that masymp t-3 graphs per group are necessary and sufficient when the per-graph signal-to-noise ratio is t<1. Signed triangle counts attain this rate, and the lower bound holds for every graph size. With aligned vertices masymp t-1 graphs suffice, so misalignment costs a factor of order t-2. When the triangle signal cancels, the rate becomes t-4 and 4-cycles are needed. Statistics built from trees have exactly the same expectation under both hypotheses, and tests based on finitely many of them have asymptotically no power. In the graphon limit, this class includes degree distributions and message-passing graph neural network features. For a non-constant null, a generic difference is visible at first order, and a simple motif test attains the aligned order of sample size, suggesting that misalignment is costly mainly for differences that are invisible at low orders. We also give an exactly valid test for one or two graphs per group, at a cost in power. In our simulations, the fitted exponents are close to the predicted ones, and degree-based and random-GNN evaluation metrics stay at their level in a setting where signed triangles need about 65 graphs.