# Network Regression with Graph Laplacians

arXiv: https://arxiv.org/abs/2109.02981 · HTML: https://mlsanity.com/paper/2109.02981/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2021.

## Built on by (most cited first, 15 of 15)

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- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
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- [Sharp Bounds for Poly-GNNs and the Effect of Graph Noise](https://mlsanity.com/paper/2407.19567/) (2024)
- [Energy Trees: Regression and Classification With Structured and Mixed-Type Covariates](https://mlsanity.com/paper/2207.04430/) (2022)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Fr'echet Geodesic Boosting](https://mlsanity.com/paper/2509.18013/) (2025)
- [Metric space valued Fr'echet regression](https://mlsanity.com/paper/2602.05225/) (2026)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Random-Effects Algorithm for Random Objects in Metric Spaces](https://mlsanity.com/paper/2605.02693/) (2026)
- [Deep Single-Index Fr'echet Regression](https://mlsanity.com/paper/2606.06957/) (2026)

## Abstract (arXiv)

Network data are increasingly available in various research fields, motivating statistical analysis for populations of networks where a network as a whole is viewed as a data point. Due to the non-Euclidean nature of networks, basic statistical tools available for scalar and vector data are no longer applicable when one aims to relate networks as outcomes to Euclidean covariates, while the study of how a network changes in dependence on covariates is often of paramount interest. This motivates to extend the notion of regression to the case of responses that are network data. Here we propose to adopt conditional Fr'echet means implemented with both global least squares regression and local weighted least squares smoothing, extending the Fr'echet regression concept to networks that are quantified by their graph Laplacians. The challenge is to characterize the space of graph Laplacians so as to justify the application of Fr'echet regression. This characterization then leads to asymptotic rates of convergence for the corresponding M-estimators by applying empirical process methods. We demonstrate the usefulness and good practical performance of the proposed framework with simulations and with network data arising from resting-state fMRI in neuroimaging, as well as New York taxi records.