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

arXiv: https://arxiv.org/abs/1903.06936 · HTML: https://mlsanity.com/paper/1903.06936/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2019.

## Built on by (most cited first, 5 of 5)

- [GraphMAD: Graph Mixup for Data Augmentation using Data-Driven Convex Clustering](https://mlsanity.com/paper/2210.15721/) (2022)
- [Joint Network Topology Inference via a Shared Graphon Model](https://mlsanity.com/paper/2209.08223/) (2022)
- [Network Estimation by Mixing: Adaptivity and More](https://mlsanity.com/paper/2106.02803/) (2021)
- [Graphon-aided Joint Estimation of Multiple Graphs](https://mlsanity.com/paper/2202.05686/) (2022)
- [Low-Complexity and Consistent Graphon Estimation from Multiple Networks](https://mlsanity.com/paper/2603.15578/) (2026)

## Abstract (arXiv)

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.