TopoFormer: Topology Meets Attention for Graph Learning

Message-passing Graph Neural Networks (GNNs) iteratively propagate and aggregate local neighborhood information followed by global readout to learn graph representations. However, their discriminative power is upper-bounded by the Weisfeiler--Lehman (1-WL) graph isomorphism test. This prevents GNNs from distinguishing certain non-isomorphic graphs with identical local neighborhood structures, often leading to similar graph representations. Unified Topological Signatures (UTS) capture compact, multi-scale representation of global graph topology derived from persistent homology. We introduce two complementary UTS signatures: Graph_UTS- a static signature of the input graph topology, and Embedding_UTS- a dynamic signature of the evolving embedding topology. They encode structural information inaccessible to 1-WL-based message-passing GNNs, yet their capabilities are explored solely for post-hoc embedding-space analysis. We integrate UTS into GNN training across three architectural interventions: (i) UTS-Aug: augmenting with standard readout feature that encodes graph's true topology; (ii) UTS-Reg: topological regularizer that constrains representation collapse; (iii) UTS-Pool: topology-guided pooling that retains structurally critical nodes. We further leverage UTS as a layer-wise diagnostic to quantify oversmoothing during GNN training. Theoretically, we show that integrating UTS into GNN optimization strictly extends GNN expressivity beyond the 1-WL hierarchy. Experiments on three graph classification benchmarks show consistent benefits: Graph-UTS, Dual-UTS, and UTS-Pool improve accuracy across all three datasets, Embedding-UTS provides smaller but similarly consistent gains, and UTS-Reg's benefit varies across graph domains. Accuracy improves by up to 5.8% with Graph-UTS augmentation, by up to 1.9% with UTS-Reg, and achieves comparable performance to TOGL with UTS-Pool.

Graph representation learning has largely focused on designing increasingly sophisticated models to transform graph topology into vector representations, or embeddings. However, the extent to which embedding quality depends on model learning, rather than on the underlying topological transformations, remains unclear. Here, we show that informative embeddings can be derived without complicated model design and gradient-based training. Propagating random features through implicit hierarchical structures induced by random walks and anonymous walks yields embeddings that capture node proximity and structural role, respectively. These two training-free embeddings preserve complementary aspects of graph organization and perform competitively with classic and recent methods across various node-, edge-, and graph-level tasks. They often require substantially less computation, resulting in a favorable quality-efficiency trade-off. Combining the two types of embeddings further improves inference quality of some tasks compared with using either embedding type alone. Our results suggest that informative graph embeddings can arise from carefully chosen topological transformations before any learning operation is applied.

Transformers have become a central architecture for graph learning, but their application to graphs requires first choosing a tokenization: a graph-to-token map that determines which structural information is exposed at the input. In this work, we show that this choice is a fundamental component of transformer expressivity. We examine three tokenizations that serve as building blocks for many existing graph tokenizations: spectral, random-walk, and adjacency tokenizations. We prove that different tokenizations induce distinct depth regimes: the same graph computation may be realizable by a shallow transformer under one tokenization, while requiring substantially larger depth under another. For example, we prove that random-walk tokenization is lossy for any walk length, making it impossible in general to recover the graph from it, and that while spectral tokenization is lossless, it is ill-conditioned for local tasks. We further show that although both random-walk and spectral tokenizations are derived from adjacency information, it is impossible for a limited-depth transformer to convert between tokenization families in general. In particular, we establish lower bounds and impossibility results showing that unfavorable tokenizations may preclude the efficient recovery of more suitable structural representations. Finally, we complement our theory with controlled experiments on synthetic and real-world tasks, validating the predicted separations and showing that different tasks favor different structural views, and combining complementary tokenizations allows the transformer to leverage distinct signals from each representation.