Eigencone Constellations on Ranked Spheres

2026-04-03

AI Summary

This work addresses the challenge of structure-preserving embedding for sparse spatial graphs and dynamic subgraph spectral distance measurement by proposing a hierarchical geometric embedding framework. Rooted graphs are mapped onto concentric spherical shells according to their distance from the root, and on each shell, star-shaped regions weighted by spectral mass are constructed to form local simplicial structures through constrained repulsive forces. The paper introduces an innovative eigencone constellation model that integrates spectral mass with spherical tiling and leverages forward deterministic isomorphism-walking trajectories to achieve efficient convergence in graph edit operations. Experiments on molecular contact maps validate the effectiveness of the proposed spectral distance and the convergence properties of the trajectories, establishing a novel paradigm for dynamic graph comparison.

Technology Category

Machine Learning: Graph-based Machine LearningKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningConstraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologiesSearch and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We introduce eigencone constellations, a hierarchical framework for embedding bounded-degree spatial graphs into concentric spherical shells and partitioning each shell into spectrally weighted, spherical star-shaped territories. Given a connected sparse spatial graph $G$ with a distinguished root vertex (the queen), we assign each vertex to a sphere whose radial position is determined by its graph distance from the queen, then tessellate each sphere into constellation territories whose solid angles are proportional to the spectral mass of the corresponding subgraph. Within each territory, nodes are packed by constrained repulsion, yielding local simplex structures. The resulting geometric representation provides a structural framework for measuring spectral distance between dynamic subgraph states. By combining this eigencone-derived metric with constraints on the domain-specific edit alphabet, we define a forward-only deterministic trajectory -- the isomorphic walk -- which converges graph edits efficiently. We define the notion of spherical star-shaped domains with geodesic visibility, establish their properties under spectral projection, and demonstrate the trajectory convergence on molecular contact graphs.

Problem

• spatial graphs
• spectral distance
• graph embedding
• dynamic subgraph states
• graph edit trajectory

Innovation

• eigencone constellations
• spectral embedding
• spherical star-shaped domains
• isomorphic walk
• graph edit trajectory