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.
We introduce Topoformer, a lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is Topo-Scan, a novel module that decomposes a graph into a short, ordered sequence of topological tokens by slicing over node or edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by a Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homology pipelines, Topo-Scan is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-art performance across graph classification and molecular property prediction benchmarks. Our results show that Topoformer matches or exceeds strong GNN and topology-based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.
Signed graphs arise in trust--distrust networks, financial correlation systems, biological interaction graphs, and many other domains in which edges can be positive or negative and may also be directed. While signed graph neural networks have improved task-specific learning, graph transfer learning on signed graphs remains underdeveloped. In this paper, we introduce TopoSIGN, a pioneer topology-guided graph pre-training and prompt learning framework for signed graphs. TopoSIGN combines a structural encoder built on the magnetic signed Laplacian with a novel persistent-homology branch that summarizes signed topology through Dowker-complex persistence images. The fused embeddings are then transferred to a prompt learning function. Experimental results on synthetic and real-world datasets demonstrate the efficacy of TopoSIGN in extracting useful structural information in signed graphs, as well as the adaptability and flexibility of the proposed general framework.
Dynamic graph neural networks (DGNNs) that operate on snapshot sequences typically fall into one of two categories. \emph{Temporal-first} approaches build per-node temporal embeddings and only afterward perform spatial aggregation, whereas \emph{Spatial-first} approaches invert this order, feeding the output of a graph convolution into a downstream temporal module. In either case, the rigid sequencing forces the second stage to consume an already-compressed summary produced by the first, ruling out joint reasoning over topology and evolution; effectively, the message-passing operator never gets to weight a neighbor's contribution by that neighbor's \emph{past} trajectory. This paper introduces \textbf{SiST-GNN} (\textbf{Si}multaneous \textbf{S}patial-\textbf{T}emporal \textbf{GNN}), which fuses the two signals inside a single message-passing operation rather than chaining them. At each snapshot, we maintain a recurrent hidden state per node that summarises its history, pairs it with the node's current feature vector, and treats the pair as two nodes joined by a cross-time edge; running a standard graph convolution on this temporally augmented graph yields the updated representation. We compare against fourteen link-prediction baselines under both the fixed-split and live-update evaluation regimes, and eleven baselines on node classification. Across the public benchmarks, SiST-GNN improves on the strongest prior method in link prediction by 1-18% in the fixed-split setting, and is the leading learned method on five of six datasets in the live-update setting, improving on the strongest prior method by 1-158% there. We additionally derive three dynamic node-classification tasks by discretizing the underlying continuous-time event streams; here SiST-GNN beats the leading discrete-time (DTDG) baseline by 7-23% and matches continuous-time (CTDG) methods that consume the raw events directly.