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Asymptotically Normal Estimation of Local Latent Network Curvature

Steven Wilkins-Reeves, Tyler McCormick

TL;DR

The paper tackles estimating the local curvature $\kappa$ of a latent network geometry from noisy distance measurements by deriving a smooth four-point estimating equation based on triangle sides and the triangle median $d_{xm}$. It introduces a clique-based distance estimator to obtain asymptotically normal distance estimates, and plugs these into a curvature estimator that yields $\hat{\kappa}$ with explicit asymptotic behavior under mild conditions. The work provides bounds for surrogate midpoints, a constrained optimization approach to enforce triangle inequalities, and a practical Newton-step variant, enabling scalable curvature inference and constant-curvature testing. Applications to cybersecurity and co-authorship networks demonstrate curvature-based anomaly detection and structural insights, with open-source code and an R package supporting implementation.

Abstract

Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent network data's complex, high-dimensional structure is to embed the graph into a low-dimensional geometric space. The curvature of this space, in particular, provides insights about the structure in the graph, such as the propensity to form triangles or present tree-like structures. We derive an estimating function for curvature based on triangle side lengths and the length of the midpoint of a side to the opposing corner. We construct an estimator where the only input is a distance matrix and also establish asymptotic normality. We next introduce a novel latent distance matrix estimator for networks and an efficient algorithm to compute the estimate via solving iterative quadratic programs. We apply this method to the Los Alamos National Laboratory Unified Network and Host dataset and show how curvature estimates can be used to detect a red-team attack faster than naive methods, as well as discover non-constant latent curvature in co-authorship networks in physics. The code for this paper is available at https://github.com/SteveJWR/netcurve, and the methods are implemented in the R package https://github.com/SteveJWR/lolaR.

Asymptotically Normal Estimation of Local Latent Network Curvature

TL;DR

The paper tackles estimating the local curvature of a latent network geometry from noisy distance measurements by deriving a smooth four-point estimating equation based on triangle sides and the triangle median . It introduces a clique-based distance estimator to obtain asymptotically normal distance estimates, and plugs these into a curvature estimator that yields with explicit asymptotic behavior under mild conditions. The work provides bounds for surrogate midpoints, a constrained optimization approach to enforce triangle inequalities, and a practical Newton-step variant, enabling scalable curvature inference and constant-curvature testing. Applications to cybersecurity and co-authorship networks demonstrate curvature-based anomaly detection and structural insights, with open-source code and an R package supporting implementation.

Abstract

Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent network data's complex, high-dimensional structure is to embed the graph into a low-dimensional geometric space. The curvature of this space, in particular, provides insights about the structure in the graph, such as the propensity to form triangles or present tree-like structures. We derive an estimating function for curvature based on triangle side lengths and the length of the midpoint of a side to the opposing corner. We construct an estimator where the only input is a distance matrix and also establish asymptotic normality. We next introduce a novel latent distance matrix estimator for networks and an efficient algorithm to compute the estimate via solving iterative quadratic programs. We apply this method to the Los Alamos National Laboratory Unified Network and Host dataset and show how curvature estimates can be used to detect a red-team attack faster than naive methods, as well as discover non-constant latent curvature in co-authorship networks in physics. The code for this paper is available at https://github.com/SteveJWR/netcurve, and the methods are implemented in the R package https://github.com/SteveJWR/lolaR.
Paper Structure (6 sections, 3 theorems, 7 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Methods
  4. Geometric Environment
  5. Identifying Curvature
  6. Estimating Curvature

Key Result

Lemma 2

If any $x,y,z \in \mathcal{M}^p(\kappa)$ where $p\geq 2$ and $x,y,z$ are not co-linear. Then $x,y,z, m \in \mathcal{M}^2(\kappa)$ where $\mathcal{M}^2(\kappa)$ is a totally geodesic submanifold of dimension 2 with constant sectional curvature $\kappa$ and $m$ is the midpoint of points $y$ and $z$.

Figures (1)

  • Figure 1: Midpoint distances and curvature of the space with equilateral triangles. The length of the triangle median $d_{xm}$ is an increasing function of the curvature $\kappa$ for fixed other triangle side lengths.

Theorems & Definitions (4)

  • Definition 1
  • Lemma 2
  • Theorem 3: Midpoint Curvature Equation
  • Theorem 4