# Graph Construction using Principal Axis Trees for Simple Graph Convolution

## TL;DR

This paper proposes a novel graph construction method using Principal Axis trees combined with supervised information to improve GNN performance, especially with simple graph convolution, by effectively creating adjacency matrices.

## Contribution

It introduces a new graph construction scheme leveraging PA-trees and supervised labels, enhancing GNNs' ability to learn from incomplete or missing adjacency information.

## Key findings

- SGC outperforms GCN in speed and results.
- Using supervised information improves graph construction.
- Careful tuning of smoothing levels prevents oversmoothing.

## Abstract

Graph Neural Networks (GNNs) are increasingly becoming the favorite method for graph learning. They exploit the semi-supervised nature of deep learning, and they bypass computational bottlenecks associated with traditional graph learning methods. In addition to the feature matrix $X$, GNNs need an adjacency matrix $A$ to perform feature propagation. In many cases, the adjacency matrix $A$ is missing. We introduce a graph construction scheme that constructs the adjacency matrix $A$ using unsupervised and supervised information. Unsupervised information characterizes the neighborhood around points. We used Principal Axis trees (PA-trees) as a source for unsupervised information, where we create edges between points falling onto the same leaf node. For supervised information, we used the concept of penalty and intrinsic graphs. A penalty graph connects points with different class labels, whereas an intrinsic graph connects points with the same class labels. We used the penalty and intrinsic graphs to remove or add edges to the graph constructed via PA-tree. We tested this graph construction scheme on two well-known GNNs: 1) Graph Convolutional Network (GCN) and 2) Simple Graph Convolution (SGC). The experiments show that it is better to use SGC because it is faster and delivers better or the same results as GCN. We also test the effect of oversmoothing on both GCN and SGC. We found out that the level of smoothing has to be carefully selected for SGC to avoid oversmoothing.