Power-law decay of the degree-sequence probabilities of multiple random graphs with application to graph isomorphism

We consider events over the probability space generated by the degree sequences of two independent Erdős-Rényi random graphs, and consider an approximation probability space where such degree sequences are deemed to be sequences of i.i.d. random variables. We show that, for any sequence of events with probabilities asymptotically smaller than some power law in the approximation model, the same upper bound also holds in the original model. We accomplish this by extending an approximation framework proposed in a seminal paper by McKay and Wormald. Finally, as an example, we apply the developed framework to bound the probability of having an isomorphism between two independent random graphs.