geometry-aware graph encoding

This paper addresses the challenge of jointly preserving local similarity and modeling global distances in graph node embedding. We propose a landmark-based shortest-path approximation method that locally preserves graph distances in low-dimensional space. Theoretically, we prove that for random graphs—including Erdős-Rényi graphs—the required embedding dimension for landmark-based representations is significantly lower than the worst-case bound dictated by Bourgain-type metric embedding theory. Empirically, graph neural networks (GNNs) efficiently learn and generalize pairwise distances between landmarks, achieving high accuracy and strong scalability on large-scale graphs. Our key contributions are twofold: (1) establishing, for the first time, a theoretical connection between random graph structure and optimal embedding dimensionality; and (2) empirically validating the strong generalization capability of GNNs in learning landmark distances. This work introduces a novel, lightweight, and scalable paradigm for graph distance modeling.

To address manifold curvature distortion induced by multi-layer implicit structures in high-dimensional multiplex graph embedding, this work is the first to model node distributions from a Riemannian geometric perspective—revealing that nodes naturally reside on highly curved non-Euclidean manifolds, with distortion intensifying as dimensionality increases. We propose a synergistic framework integrating hierarchical dimensional embedding and a hyperbolic graph neural network (Hyperbolic GNN): the former progressively learns compact, expressive latent dimensions, while the latter explicitly encodes negative curvature in hyperbolic space; both components are jointly optimized for Gaussian curvature-aware embedding. Evaluated on real-world high-dimensional multiplex graphs, our method significantly reduces geometric distortion and consistently outperforms state-of-the-art approaches across downstream tasks—including link prediction and node classification.

Existing graph positional encodings (PEs) frequently fail—or even degrade GNN performance—on heterogeneous graphs (where neighboring nodes exhibit large label disparities), despite the ubiquity of heterogeneity in real-world networks. To address this, we propose Learnable Laplacian Positional Encoding (LLPE), the first PE framework theoretically and empirically tailored to heterogeneous graphs. LLPE leverages the full spectral decomposition of the graph Laplacian and introduces learnable frequency-domain filters to jointly model both homophilous and heterophilous structural patterns. Crucially, it supports arbitrary graph distance approximation, thereby breaking the fundamental reliance of conventional PEs on homophily assumptions. LLPE integrates seamlessly into both GNNs and Graph Transformers. Evaluated on 12 benchmark datasets, it yields substantial improvements: up to 35% accuracy gain on synthetic graphs and up to 14% on real-world graphs.

Existing methods for generating manifold meshes typically rely on indirect representations—such as level sets or template deformations—making it difficult to directly produce high-quality, topologically unconstrained polygonal meshes with structural integrity. This paper introduces the first end-to-end differentiable framework that explicitly models half-edge structure via vertex-level continuous connectivity embeddings, enabling direct generation of discrete manifold-conforming meshes in a continuous latent space. Key contributions include: (1) the first continuous neighborhood relation learning mechanism; (2) mesh distribution fitting via stochastic optimization; and (3) topology-agnostic generation and repair capabilities. Evaluated on large-scale datasets, our method significantly improves mesh element quality, geometric fidelity, and topological diversity. It establishes the first truly end-to-end differentiable approach for manifold mesh generation and repair, bridging a critical gap between implicit representation learning and explicit, valid mesh synthesis.

This study addresses the lack of systematic evaluation of hyperbolic graph embedding methods for link prediction and topological reconstruction. For the first time, it conducts a cross-disciplinary benchmark of thirteen unsupervised hyperbolic embedding approaches—drawn from machine learning, network science, and algorithms—within a unified experimental framework, encompassing maximum likelihood estimation, representation learning, and hybrid paradigms. The results reveal that performance differences stem primarily from the embedding paradigm rather than disciplinary origin, with maximum likelihood and representation learning methods generally outperforming others. However, no single method universally excels across all network structures and tasks. This work provides practical guidance for method selection and clarifies the network contexts in which each approach is most effective.

This work addresses the challenge of efficient lossless compression for large-scale real-world graph data by proposing a novel algorithm that leverages geometric representations of graph structure through direct application of modern hyperbolic space embeddings. By capitalizing on the intrinsic hyperbolic geometry inherent in complex networks, the method achieves substantially improved compression efficiency while preserving lossless reconstruction. Experimental evaluation across diverse real-world graph datasets demonstrates that the proposed approach outperforms the current state-of-the-art methods by up to 42% in compression ratio, thereby validating the efficacy and superiority of hyperbolic embeddings for graph compression tasks.

This work addresses the challenge of effectively integrating multi-scale topological structures to enhance graph representation learning. The authors propose TopoFormer, a novel framework featuring a parallelizable and persistence-free Topo-Scan module that converts graph structures into ordered topological sequences, which are then seamlessly integrated into a Transformer architecture. This design enables unified modeling of topological patterns ranging from local to global scales. By synergistically combining topological data analysis, node/edge filtering, and sequential encoding, TopoFormer achieves competitive or superior performance compared to existing graph neural networks and topological methods on graph classification and molecular property prediction tasks, while maintaining efficient and predictable computational overhead.

This work addresses the limitations of existing representation alignment methods, which predominantly rely on geometric properties and struggle to capture the global structural organization of model representations. To overcome this, the study introduces topological data analysis into the field for the first time, proposing a Mapper-based visual analytics framework. By integrating force-directed layout, Bubble Sets, motif querying, and membrane-inspired heuristics, the framework enables a unified analytical pipeline spanning global structure alignment, local region matching, and fine-grained pattern exploration. Case studies on language and multimodal models, complemented by expert evaluations, demonstrate that the approach effectively reveals and compares the topological organization of representations across different models or layers, offering deep structural insights.