# EM-Based Smooth Graphon Estimation Using Bayesian and Spline-Based Approaches

## TL;DR

This paper introduces an EM-based method for smooth graphon estimation that combines Bayesian inference and spline regression, allowing for flexible node ordering and uncertainty quantification in network data analysis.

## Contribution

It presents a novel EM algorithm integrating Bayesian and spline approaches for nonparametric, smooth graphon estimation without requiring monotonicity constraints.

## Key findings

- Effective in capturing smooth graphons from network data
- Allows comparison of different node ordering strategies
- Demonstrates robustness through simulations and examples

## 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 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.