Topology-Selected Manifold Routing for Multi-Relational Message Passing
DOI:
https://doi.org/10.66238/smsrj.v4i2.74Keywords:
Multi-relational Graphs, Manifold Learning, Message Passing, Hyperbolic Geometry, Dynamic RoutingAbstract
Multi-relational graphs present a persistent challenge in representation learning due to their complex hierarchical structures and heterogeneous edge types. Standard message passing neural networks fundamentally operate in Euclidean space, which is characterized by zero curvature and isotropic capacity. However, multi-relational networks typically exhibit varying topological characteristics across different relational subgraphs, ranging from tree-like hierarchical dependencies to dense cyclic communities. Embedding such diverse topologies into a single constant curvature space leads to severe structural distortion and feature crushing, especially as the number of message passing layers increases. This paper introduces Topology-Selected Manifold Routing, a novel architectural paradigm for multi-relational message passing that addresses these structural bottlenecks. By conceptualizing the embedding space as a product of heterogeneous manifolds including hyperbolic, spherical, and Euclidean geometries, our methodology dynamically routes information across the most suitable geometric spaces tailored to the localized topology of each relation. The proposed framework computes relational curvature profiles using discrete Ricci curvature approximations and employs a manifold-aware routing mechanism to aggregate relational messages. We establish the theoretical foundations of mixed-curvature message passing and provide a rigorous analysis of its convergence properties under dynamic routing. Through extensive evaluations on standard benchmark multi-relational graphs, we demonstrate that Topology-Selected Manifold Routing significantly mitigates over-smoothing, preserves complex hierarchical motifs, and achieves superior expressive power compared to existing homogeneous space models.
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