A Geometric Perspective for High-Dimensional Multiplex Graphs
Kamel Abdous, Nairouz Mrabah, Mohamed Bouguessa
TL;DR
The paper tackles the challenge of embedding high-dimensional multiplex graphs, where many dimensions introduce geometric distortions that push node representations onto highly curved manifolds. It introduces HYPER-MGE, a method that combines hierarchical dimension aggregation with hyperbolic embeddings (Poincaré/Lorentz) to produce flat, low-dimensional latent spaces, mitigating distortions while capturing both intra- and inter-dimension hierarchy. The study provides a geometric analysis of latent-space curvature via Intrinsic Dimension ($ID$) and Linear Intrinsic Dimension ($LID$), demonstrates distortions grow with the number of dimensions, and shows that the proposed approach reduces this curvature, yielding improved performance on link prediction and node classification across real-world datasets. Empirical results and ablations indicate that the synergy between hierarchical aggregations and hyperbolic embeddings is crucial for effectively decoding the complex structure of high-dimensional multiplex graphs and enabling robust downstream inference.
Abstract
High-dimensional multiplex graphs are characterized by their high number of complementary and divergent dimensions. The existence of multiple hierarchical latent relations between the graph dimensions poses significant challenges to embedding methods. In particular, the geometric distortions that might occur in the representational space have been overlooked in the literature. This work studies the problem of high-dimensional multiplex graph embedding from a geometric perspective. We find that the node representations reside on highly curved manifolds, thus rendering their exploitation more challenging for downstream tasks. Moreover, our study reveals that increasing the number of graph dimensions can cause further distortions to the highly curved manifolds. To address this problem, we propose a novel multiplex graph embedding method that harnesses hierarchical dimension embedding and Hyperbolic Graph Neural Networks. The proposed approach hierarchically extracts hyperbolic node representations that reside on Riemannian manifolds while gradually learning fewer and more expressive latent dimensions of the multiplex graph. Experimental results on real-world high-dimensional multiplex graphs show that the synergy between hierarchical and hyperbolic embeddings incurs much fewer geometric distortions and brings notable improvements over state-of-the-art approaches on downstream tasks.