  This paper proposes the estimation of a smooth graphon model for network data
analysis using principles of the EM algorithm. The approach considers both
variability with respect to ordering the nodes of a network and smooth
estimation of the graphon by nonparametric regression. To do so, (linear)
B-splines are used, which allow for smooth estimation of the graphon,
conditional on the node ordering. This provides the M-step. The true ordering
of the nodes arising from the graphon model remains unobserved and Bayesian
ideas are employed to obtain posterior samples given the network data. This
yields the E-step. Combining both steps gives an EM-based approach for smooth
graphon estimation. Unlike common other methods, this procedure does not
require the restriction of a monotonic marginal function. The proposed graphon
estimate allows to explore node-ordering strategies and therefore to compare
the common degree-based node ranking with the ordering conditional on the
network. Variability and uncertainty are taken into account using MCMC
techniques. Examples and simulation studies support the applicability of the
approach.
