Hyperbolic embedding of multilayer networks

TL;DR

A novel hyperbolic embedding framework is introduced that generates layer-specific hyperbolic embeddings, enabling detailed intra-layer analysis and inter-layer comparisons, while simultaneously preserving the global multilayer structure within hyperbolic space.

Abstract

Multilayer networks offer a powerful framework for modeling complex systems across diverse domains, effectively capturing multiple types of connections and interdependent subsystems commonly found in real world scenarios. To analyze these networks, embedding techniques that project nodes into a lower-dimensional geometric space are essential. This paper introduces a novel hyperbolic embedding framework that advances the state of the art in multilayer network analysis. Our method, which supports heterogeneous node sets across networks and inter-layer connections, generates layer-specific hyperbolic embeddings, enabling detailed intra-layer analysis and inter-layer comparisons, while simultaneously preserving the global multilayer structure within hyperbolic space, a capability that sets it apart from existing approaches, which typically rely on independent embedding of layers. Through experiments on synthetic multilayer stochastic block models, we demonstrate that our approach effectively preserves community structure, even when layers consist of different node sets. When applied to real brain networks, the method successfully clusters disease-related brain regions from different patients, outperforming layer-independent approaches and highlighting its relevance for comparative analysis. Overall, this work provides a robust tool for multilayer network analysis, enhancing interpretability and offering new insights into the structure and function of complex systems.

Figures (4)

• Figure 1: Global pipeline for multiple network embedding with identical nodes set across layers. The process begins with the extraction of the connectivity matrices for each layer. These matrices are then combined to form a global connectivity matrix $G$. The dimension reduction algorithm is applied to $G$ to obtain a two-dimensional representation of the dataset. From this embedding, the angular coordinates of the nodes in each layer are extracted, and their radii are initially normalized to one. In a final step, the radial coordinates are reassigned based on the centrality of each node.
• Figure 2: Hyperbolic embeddings of multilayer SBM models. Parameters: 3 layers, 3 communities; $p_{=} = 0.16$, $p_{\neq} = 0.06$, $\mu = 20$, $\alpha = 10$. (A) Comparison between multilayer and independent embeddings for identical node sets across layers. Each layer contains 100 nodes. Top row: embeddings computed using the proposed multilayer approach. Bottom row: embeddings computed independently for each layer. (B) Multilayer embeddings with varying node sets across layers. The multilayer model comprises 3 layers with different numbers of nodes (Layer 0: 100 nodes; Layer 1: 90 nodes; Layer 2: 85 nodes).
• Figure 3: Hyperbolic embedding of patients with left temporal lobe epilepsy (red points) and controls (blue points). Visualization of the Gaussian distributions of node positions corresponding to the left temporal lobes (2 nodes per patients) within the Poincaré disk, using two embedding strategies: A) the proposed multilayer embedding approach, where each layer represents an individual patient $\beta = 50, \mu = 20$; and B) an independent embedding of each patient followed by post hoc rotational alignment.
• Figure 4: $G_{\text{score}}$ (See Eq. \ref{['eq_g_Score']}) as a function of the coupling parameter $\mu$ and the mean edge weight of the graph. Calculations were performed on a two-layer SBM model with $n = 100$ nodes per layer.