Identifiability of Latent Space Network Models on Anisotropic Thurston Geometries

2026-09-07

AI Summary

研究解决了在非恒定曲率和非乘积的三维模型几何中，网络如何识别位置及检测几何的问题，使用了Heisenberg群等模型并通过分析有向无向连接来探讨。

Abstract

A latent space network model places the nodes in a metric space and lets the probability of a tie decrease with distance. In a space of constant curvature, pairwise distances determine the positions up to an isometry. In the products and in the three remaining three-dimensional model geometries they do not. We study the three geometries that are neither of constant curvature nor products: the Heisenberg group, the solvable group and the universal cover of the unit tangent bundle of the hyperbolic plane. We ask what one network identifies about positions in them and when their geometry is detectable. Two anchors remove the isometry ambiguity. Small configurations are not determined by their distances, and generic local identification holds beyond a finite threshold, certified in the Heisenberg group. We derive the posterior on the quotient by the isometry group. For small configurations, the divergence to the nearest product or constant-curvature competitor is the stress component of the curvature difference and vanishes at high order in the scale; undirected ties therefore detect the geometry only in large networks with large enclosed areas. Directed ties expose it at first order: in the two twisted geometries, asymmetric preferences circulate around triangles in proportion to enclosed area, which no additive ranking produces. On dense competitive-game counter networks, the coupled model beats rankings on every geometry, degree-corrected rankings and free antisymmetric terms. A Euclidean model with the same coupled term matches it, so the gain is the coupling of similarity and circulation through shared coordinates. An additive-and-multiplicative-effects model predicts better still.

Problem

• latent space network model
• anisotropic Thurston geometries
• identifiability
• Heisenberg group
• solvable group

Innovation

• latent space network model
• anisotropic Thurston geometries
• isometry ambiguity
• generic local identification
• coupled model