EM-based smooth graphon estimation using MCMC and spline-based approaches

Abstract

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 MCMC techniques are employed to obtain position samples conditional on the network. 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 relying on the MCMC sequences. Examples and simulation studies support the applicability of the approach.