Non-backtracking Graph Neural Networks

TL;DR

This paper tackles the backtracking-induced redundancies in standard GNNs, which amplify message-flow counts and cause over-squashing. It introduces NBA-GNN, a non-backtracking graph neural network that assigns and updates edge-directed features $h_{i\rightarrow j}^{(t)}$ without reusing reverse messages, and analyzes its impact through sensitivity bounds and SBM expressivity. The authors prove that NBA-GNN yields larger sensitivity bounds than conventional GNNs, enabling more faithful propagation in sparse graphs, and demonstrate the ability to recover SBM structure with average degree $\omega(1)$ and $n^{o(1)}$ even under sparse conditions. Empirically, NBA-GNN achieves competitive or state-of-the-art results on the Long-Range Graph Benchmark and consistently improves transductive node classification across diverse datasets and backbones, with ablations underscoring the essential role of non-backtracking updates. Overall, the work bridges non-backtracking spectral concepts with end-to-end GNN design, offering a scalable approach to mitigate over-squashing and enhance performance on long-range and sparse graph tasks.

Abstract

The celebrated message-passing updates for graph neural networks allow representing large-scale graphs with local and computationally tractable updates. However, the updates suffer from backtracking, i.e., a message flowing through the same edge twice and revisiting the previously visited node. Since the number of message flows increases exponentially with the number of updates, the redundancy in local updates prevents the graph neural network from accurately recognizing a particular message flow relevant for downstream tasks. In this work, we propose to resolve such a redundancy issue via the non-backtracking graph neural network (NBA-GNN) that updates a message without incorporating the message from the previously visited node. We theoretically investigate how NBA-GNN alleviates the over-squashing of GNNs, and establish a connection between NBA-GNN and the impressive performance of non-backtracking updates for stochastic block model recovery. Furthermore, we empirically verify the effectiveness of our NBA-GNN on the long-range graph benchmark and transductive node classification problems.

Figures (5)

• Figure 1: Comparison of two types of message flow. Step 1: Message flow from node $i$ to node $j$. Step 2: The simple update includes the message flow from node $j$ back to node $i$ (left) at step 1, while the non-backtracking update removes this redundant message flow (right).
• Figure 2: Two non-backtracking walks (right) are sufficient to express information contained in a walk with backtracking transition (left).
• Figure 3: Computation graph of typical GNN and NBA-GNN predicting node "$0$". (\ref{['subfig:comp_graph_typ']}) Redundant messages increase the size of the computation graph, proportional to the number of layers. (\ref{['subfig:comp_graph_nba']}) NBA-GNN assigns a pair of features for each edge and updates them via non-backtracking message passing. By reducing redundant messages, it results in a simplified computation graph compared to typical GNNs.
• Figure 4: Ablation studies on the components of NBA-GNN.
• Figure 5: References for section \ref{['subsec:cond-at']}

Theorems & Definitions (37)

• Proposition 1
• Proposition 2: Sensitivity bounds
• Lemma 1: Sensitivity bounds of NBA-GNNs
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 5
• Lemma 2
• proof
• Lemma 3