# Modeling Tree-like Heterophily on Symmetric Matrix Manifolds

## TL;DR

This paper introduces a new method for modeling complex network structures using symmetric matrix manifolds to better handle diverse substructures in real-world networks.

## Contribution

The novelty lies in using symmetric positive-definite matrix manifolds for graph neural networks to model tree-like heterophily.

## Key findings

- The proposed method outperforms Euclidean and hyperbolic geometry-based models in semi-supervised node classification.
- Symmetric manifolds provide a more effective geometric space for modeling mixed network substructures.
- Riemannian metrics enhance information propagation in the proposed graph neural network.

## Abstract

Tree-like structures, characterized by hierarchical relationships and power-law distributions, are prevalent in a multitude of real-world networks, ranging from social networks to citation networks and protein–protein interaction networks. Recently, there has been significant interest in utilizing hyperbolic space to model these structures, owing to its capability to represent them with diminished distortions compared to flat Euclidean space. However, real-world networks often display a blend of flat, tree-like, and circular substructures, resulting in heterophily. To address this diversity of substructures, this study aims to investigate the reconstruction of graph neural networks on the symmetric manifold, which offers a comprehensive geometric space for more effective modeling of tree-like heterophily. To achieve this objective, we propose a graph convolutional neural network operating on the symmetric positive-definite matrix manifold, leveraging Riemannian metrics to facilitate the scheme of information propagation. Extensive experiments conducted on semi-supervised node classification tasks validate the superiority of the proposed approach, demonstrating that it outperforms comparative models based on Euclidean and hyperbolic geometries.