Dirac--Bianconi Graph Neural Networks -- Enabling Non-Diffusive Long-Range Graph Predictions

TL;DR

The paper tackles the limitation of diffusion-dominated GNNs by introducing Dirac--Bianconi GNNs (DBGNNs) that leverage the topological Dirac equation to enable non-diffusive, wave-like long-range propagation on graphs. It defines the Dirac--Bianconi layer (DBT) and the DB 1-Step (DB1S) building blocks, stacking them with shared weights and skip connections to form the DBGNN, and demonstrates that this architecture preserves feature heterogeneity (nonzero Dirichlet energy) across depth. Empirically, DBGNNs achieve state-of-the-art or competitive performance on long-range tasks such as power-grid stability and peptide property prediction, requiring fewer parameters than some baselines and showing strong out-of-distribution generalization. The work provides a principled, physics-inspired alternative to diffusion-based MPNNs, highlighting long-range graph dynamics and potential extensions to incorporate attention-like mechanisms.

Abstract

The geometry of a graph is encoded in dynamical processes on the graph. Many graph neural network (GNN) architectures are inspired by such dynamical systems, typically based on the graph Laplacian. Here, we introduce Dirac--Bianconi GNNs (DBGNNs), which are based on the topological Dirac equation recently proposed by Bianconi. Based on the graph Laplacian, we demonstrate that DBGNNs explore the geometry of the graph in a fundamentally different way than conventional message passing neural networks (MPNNs). While regular MPNNs propagate features diffusively, analogous to the heat equation, DBGNNs allow for coherent long-range propagation. Experimental results showcase the superior performance of DBGNNs over existing conventional MPNNs for long-range predictions of power grid stability and peptide properties. This study highlights the effectiveness of DBGNNs in capturing intricate graph dynamics, providing notable advancements in GNN architectures.

Figures (7)

• Figure 1: Feature activation versus steps of linear DB \ref{['eq:linDB']} (left) and MPNN \ref{['eq:MPNN']} (right) on a path graph, $d_n = d_f = 1$, same random weights. The initial condition has all features zero, except at node 1, where the node features are randomly activated. Linear DB shows activation moving linearly down the graph, while MPNN shows diffusion.
• Figure 2: Illustration of the Dirac--Bianconi Graph Neural Network (DBGNN) layer (c) and its components. The Dirac--Bianconi T-step layer (b) consists of multiple DB1S (a). By stacking multiple DB1S and applying them sequentially, information can be propagated along the graph.
• Figure 3: Evolution of the normalized Dirichlet energy (DE) of the node feature embeddings for a sample of dataset20 with five different seeds and no training. For DBGNNs, the DE remains at a high level, meaning that information can be deeply propagated, while for GCNs, information is quickly lost.
• Figure 4: Oscillatory regime: Feature activation versus steps of the linear DB \ref{['eq:linDB']} (top left), the non-linear DB 1-step layer \ref{['eq:linDB']} + ReLU (top right), the linear MPNN layer \ref{['eq:MPNN']} (bottom left), and MPNN \ref{['eq:MPNN-sigma']} without (middle) and with non-linear messages (bottom right). $d_n = d_f = 4$, same random weights. In the case of DBGNN, information can be propagated and is not lost by diffusion.
• Figure 5: Evolution of normalized Dirichlet energy of node feature embeddings in a trained DBGNN layer for a sample of dataset20 with five different seeds.