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Hyperbolic Network Latent Space Model with Learnable Curvature

Jinming Li, Gongjun Xu, Ji Zhu

TL;DR

This work develops a hyperbolic latent space model with a learnable curvature parameter $-K<0$ for network data, arguing that curvature critically shapes embedding fidelity and cannot be replaced by latent-position distributions alone. It provides a matrix-form distance representation $\Theta = \frac{1}{\sqrt{K}} \operatorname{acosh}(-Z\Lambda Z^T)$ and a gradient-based estimation procedure on the hyperboloid manifold, with initialization via USVT and theoretical guarantees including embedding-error bounds and consistency rates: $\frac{1}{n(n-1)}\|\hat{P}-P\|_F^2, \frac{1}{n(n-1)}\|\hat{\Theta}-\Theta\|_F^2 = O_p(\sqrt{(d+1)/n})$, and, for $d=2$, $|\hat{K}-K|^2=O_p(1/\sqrt{n})$, $\|\hat{Z}-Z\|_F^2/n=O_p(1/\sqrt{n})$. Through simulations and a real-world Facebook network, the authors demonstrate that learning $K$ improves both model fit and downstream tasks (e.g., link prediction) and reveal clear effects of curvature on sparsity and community-structured embeddings. The results justify incorporating latent-space curvature into network analysis and point to practical extensions such as covariate integration and non-constant curvature.

Abstract

Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a significant, unresolved issue. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter. We theoretically justify that learning the optimal curvature is essential to minimizing the embedding error across all hyperbolic embedding methods beyond network latent space models. A maximum-likelihood estimation strategy, employing manifold gradient optimization, is developed, and we establish the consistency and convergence rates for the maximum-likelihood estimators, both of which are technically challenging due to the non-linearity and non-convexity of the hyperbolic distance metric. We further demonstrate the geometric effect of latent space curvature and the superior performance of the proposed model through extensive simulation studies and an application using a Facebook friendship network.

Hyperbolic Network Latent Space Model with Learnable Curvature

TL;DR

This work develops a hyperbolic latent space model with a learnable curvature parameter for network data, arguing that curvature critically shapes embedding fidelity and cannot be replaced by latent-position distributions alone. It provides a matrix-form distance representation and a gradient-based estimation procedure on the hyperboloid manifold, with initialization via USVT and theoretical guarantees including embedding-error bounds and consistency rates: , and, for , , . Through simulations and a real-world Facebook network, the authors demonstrate that learning improves both model fit and downstream tasks (e.g., link prediction) and reveal clear effects of curvature on sparsity and community-structured embeddings. The results justify incorporating latent-space curvature into network analysis and point to practical extensions such as covariate integration and non-constant curvature.

Abstract

Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a significant, unresolved issue. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter. We theoretically justify that learning the optimal curvature is essential to minimizing the embedding error across all hyperbolic embedding methods beyond network latent space models. A maximum-likelihood estimation strategy, employing manifold gradient optimization, is developed, and we establish the consistency and convergence rates for the maximum-likelihood estimators, both of which are technically challenging due to the non-linearity and non-convexity of the hyperbolic distance metric. We further demonstrate the geometric effect of latent space curvature and the superior performance of the proposed model through extensive simulation studies and an application using a Facebook friendship network.
Paper Structure (19 sections, 3 theorems, 11 equations, 9 figures, 1 table, 3 algorithms)

This paper contains 19 sections, 3 theorems, 11 equations, 9 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Hyperbolic Network Latent Space Model
  3. Preliminary on Hyperbolic Geometry
  4. Poincaré Disk Model
  5. Hyperboloid Model
  6. Model Setup
  7. Identifiability
  8. Estimation Procedure
  9. Theoretical Analysis
  10. Embedding Error Caused by Misspecified Curvature
  11. Consistency Results
  12. Simulation Studies
  13. Model Properties versus Latent Space Curvature
  14. Consistency of the Maximum-Likelihood Estimators
  15. Embedding Error versus Latent Space Curvature
  16. ...and 4 more sections

Key Result

Theorem 4.1

Consider (1) two hyperbolic spaces $\mathbb{H}_2^{-K}$ and $\mathbb{H}_2^{-K'}$ with $K \neq K'$, and (2) a set of discrete points $\{z_i\}_{i = 1}^n \subset \mathbb{H}_2^{-K}$. Suppose Assumption I holds and additionally, the following smoothness assumption holds: (1) when $K<K'$, there exists a ho

Figures (9)

  • Figure 1: An example of embedding tree-structured data. Left: embedded with $2$-dimensional Euclidean space; Right: embedded with $2$-dimensional Hyperbolic space.
  • Figure 2: Left: Geodesics of $OA$ and $OB$ on the Poincaré disk model; Right: corresponding hyperbolic and Euclidean distances for OA and OB.
  • Figure 3: Left: a triangle on the Poincaré disk model; Middle: isometry illustration from hyperboloid model to Poincaré disk model; Right: a triangle on the hyperboloid model.
  • Figure 4: A toy network with visualization of latent positions and distance heatmap.
  • Figure 5: Log-scaled relative errors of maximum-likelihood estimators.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.2: Consistency of $\hat{\Theta}$ and $\hat{P}$
  • Theorem 4.3: Consistency of $\hat{K}$ and $\hat{Z}$
  • Remark 4.3
  • Remark 4.4