This paper focuses on the comparison of networks on the basis of statistical inference. For that purpose, we rely on smooth graphon models as a nonparametric modeling strategy that is able to capture complex structural patterns. The graphon itself can be viewed more broadly as local density or intensity function on networks, making the model a natural choice for comparison purposes. More precisely, to gain information about the (dis-)similarity between networks, we extend graphon estimation towards modeling multiple networks simultaneously. In particular, fitting a single model implies aligning different networks with respect to the same graphon estimate. To do so, we employ an EM-type algorithm. Drawing on this network alignment consequently allows a comparison of the edge density at local level. Based on that, we construct a chi-squared-type test on equivalence of network structures. Simulation studies and real-world examples support the applicability of our network comparison strategy.
