Aggregating Direct and Indirect Neighbors through Graph Linear Transformations

TL;DR

This paper introduces Graph Linear Transformations, a novel linear operator derived from graph structures, enabling nodes to aggregate information from both direct and indirect neighbors efficiently, improving over traditional GNNs.

Contribution

It proposes a new linear transformation based on Gaussian Belief Propagation that captures multi-hop information without explicit path enumeration, offering interpretable propagation biases.

Findings

Achieves competitive or superior performance on benchmark datasets.

Enables direct and indirect neighbor aggregation with a single linear operator.

Provides interpretable insights into propagation biases through different precision matrix constructions.

Abstract

Graph neural networks (GNN) typically rely on localized message passing, requiring increasing depth to capture long range dependencies. In this work, we introduce Graph Linear Transformations, a linear transformation that realizes direct and indirect feature mixing on graphs through a single, well-defined linear operator derived from the graph structure. By interpreting graphs as walk-summable Gaussian graphical models, we compute these transformations via Gaussian Belief Propagation, enabling each node to aggregate information from both direct and indirect neighbors without explicit enumeration of multi-hop paths. We show that different constructions of the underlying precision matrix induce distinct and interpretable propagation biases, ranging from selective edge-level interactions to uniform structural smoothing, and that Graph Linear Transformations can achieve competitive or superior performance compared to both local message-passing GNNs and dynamic neighborhood aggregation models across homophilic and heterophilic benchmark datasets.