Commute Graph Neural Networks

TL;DR

This work targets the challenge of capturing asymmetric, mutual path dependencies in directed graphs, which standard GNNs and many DirGNNs fail to model effectively. It introduces Commute Graph Neural Networks (CGNN), anchored by a novel digraph Laplacian (DiLap) that enables efficient computation of commute times via a sparse, SVD-based pseudoinverse, and integrates these commute-time proximities into the message-passing scheme. The method also includes a similarity-based graph rewiring step to ensure irreducibility and aperiodicity without sacrificing sparsity, and defines per-layer updates that weight neighbor messages by direction-aware commute-time affinities. Empirical results on eight digraph benchmarks show CGNN achieving state-of-the-art accuracy on six datasets and robust performance on both homophilic and heterophilic graphs, highlighting the practical impact of leveraging random-walk distances in directed graph learning. The work also analyzes complexity, ablations, and design choices (rewiring vs PPR, directionality, SVD rank) to provide guidance on when and how commute-time information benefits real-world graphs. A key limitation is memory overhead from the dense commute-time matrix, suggesting future work on local, scalable approximations.

Abstract

Graph Neural Networks (GNNs) have shown remarkable success in learning from graph-structured data. However, their application to directed graphs (digraphs) presents unique challenges, primarily due to the inherent asymmetry in node relationships. Traditional GNNs are adept at capturing unidirectional relations but fall short in encoding the mutual path dependencies between nodes, such as asymmetrical shortest paths typically found in digraphs. Recognizing this gap, we introduce Commute Graph Neural Networks (CGNN), an approach that seamlessly integrates node-wise commute time into the message passing scheme. The cornerstone of CGNN is an efficient method for computing commute time using a newly formulated digraph Laplacian. Commute time is then integrated into the neighborhood aggregation process, with neighbor contributions weighted according to their respective commute time to the central node in each layer. It enables CGNN to directly capture the mutual, asymmetric relationships in digraphs. Extensive experiments on 8 benchmarking datasets confirm the superiority of CGNN against 13 state-of-the-art methods.

Figures (5)

• Figure 1: A digraph and its undirected counterpart. Blue arrows indicate unidirectional paths, together with longer paths in the gray area, forming commute closed loops between the central node $v_i$ and its outgoing neighbors $v_j$ and $v_k$. In the undirected graph, shortest path distances (SPD) between nodes are symmetric. However, in the digraph, the fact that unidirectional SPDs are equal does not imply that mutual SPDs will also be equal. For instance, while the SPDs from $v_i$ to $v_j$ and $v_k$ are identical, the reverse SPD from $v_j$ and $v_k$ back to $v_i$ do not necessarily match these distances.
• Figure 2: The sorted node indices in $\Omega$ are connected one by one with undirected edges to construct $G^\prime$, then adding all edges from $G^\prime$ to $G$ to generate $\widetilde{G}$.
• Figure 3: Distance between $\mathcal{M}$ and $\mathbf{A}$, and between $\mathcal{M}$ and $\widetilde{\mathcal{C}}$.
• Figure 4: Accuracy vs. running time.
• Figure 5: Sensitivity analysis on Squirrel and CoraML.

Theorems & Definitions (7)

• Proposition 3.2
• Theorem 3.3
• Proposition 4.1
• Lemma 4.2
• Theorem 4.3
• proof
• proof