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.

翻译：本文建议使用EM算法的原则对网络数据分析的平滑图形模型进行估算。 这种方法既考虑网络节点的订购的变异性,又考虑以非参数回归法对图形进行平滑估算。 为此,使用(线性)B- splines,允许以节点顺序为条件对图形on进行平稳估算。 这提供了M级。 由图形模型产生的节点的真正排序仍然不为人知,巴耶西亚人的想法用于根据网络数据获取后方样本。 这产生了E级。 将两个步骤结合起来,为平滑图形估算提供了基于EM的方法。 与其他常见方法不同,这一程序不需要限制单调边际功能。 拟议的图形估计允许探索无线排序战略,从而将普通的基于度的节点排序与网络的排序进行比较。 采用MCMC技术将易变性和不确定性考虑在内。 实例和模拟研究支持该方法的可适用性。