Heterogeneous manifolds for curvature-aware graph embedding

Abstract

Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space matches that of the graph. Euclidean spaces are often a poor choice for many types of real-world graphs, where hierarchical structure and a power-law degree distribution are linked to negative curvature. In this regard, it has recently been shown that hyperbolic spaces and more general manifolds, such as products of constant-curvature spaces and matrix manifolds, are advantageous to approximately match nodes pairwise distances. However, all these classes of manifolds are homogeneous, implying that the curvature distribution is the same at each point, making them unsuited to match the local curvature (and related structural properties) of the graph. In this paper, we study graph embeddings in a broader class of heterogeneous rotationally-symmetric manifolds. By adding a single extra radial dimension to any given existing homogeneous model, we can both account for heterogeneous curvature distributions on graphs and pairwise distances. We evaluate our approach on reconstruction tasks on synthetic and real datasets and show its potential in better preservation of high-order structures and heterogeneous random graphs generation.

Hyperbolic graph embedding of MEG brain networks to study brain alterations in individuals with subjective cognitive decline

“…This is because brain networks are scale-free graphs with a tree-like hierarchical structure, where the number of nodes grows exponentially as we move away from higher towards lower hierarchy regions. This substantially exceeds the polynomial expanding space capacity of the Euclidean space, resulting in embeddings with high distortion [25], [26]. To mitigate this problem, a viable solution is to increase the dimension of the Euclidean embedding space.…”