  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 density or intensity
function on networks, making the model a natural choice for comparison
purposes. Extending graphon estimation towards modeling multiple networks
simultaneously consequently provides substantial information about the
(dis-)similarity between networks. Fitting such a joint model - which can be
accomplished by applying an EM-type algorithm - provides a joint graphon
estimate plus a corresponding prediction of the node positions for each
network. In particular, it entails a generalized network alignment, where
nearby nodes play similar structural roles in their respective domains. Given
that, we construct a chi-squared test on equivalence of network structures.
Simulation studies and real-world examples support the applicability of our
network comparison strategy.
