# Consistent response prediction for multilayer networks on unknown manifolds

**Authors:** Aranyak Acharyya, Jesús Arroyo Relión, Michael Clayton, Marta Zlatic, Youngser Park, Carey E. Priebe  
**arXiv:** [2405.03225](https://arxiv.org/abs/2405.03225)  
**Date:** 2024-05-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2405.03225v1/

## Key points

- Algorithm for predicting responses in networks with common nodes
- Utilizes common subspace independent edge model
- Captures heterogeneity among networks through low dimensional matrices
- Uses isomap to learn underlying manifold and predict responses consistently
- Theoretical justifications supported by numerical simulations
- Applied to larval Drosophila connectome data
- Explores finite-sample improvements for scenarios with limited graph availability
- Acknowledges the challenge of higher-dimensional manifolds as an open problem for future research

## Summary

Our paper presents an algorithm for predicting responses in a collection of networks that share a common set of nodes. These networks are associated with responses and represent points on a one-dimensional manifold within a higher dimensional space. Our proposed algorithm utilizes the common subspace independent edge model to consistently predict the response for an unlabeled network. This model captures the heterogeneity among networks through low dimensional matrices while sharing a common invariant subspace. By estimating these matrices and learning the underlying manifold using isomap, our algorithm can predict responses at unlabeled networks with consistency. The theoretical justifications for our algorithm are supported by numerical simulations. We also extend our research by applying the algorithm to larval Drosophila connectome data and explore finite-sample improvements for scenarios with limited graph availability. However, we acknowledge that the underlying manifold may have dimensions higher than one and consider this as an open problem for future research. Overall, our work provides insights into predicting responses in multilayer networks on unknown manifolds and offers potential avenues for further exploration in refining the algorithm's performance in real-life applications with limited data availability.