Identifiability of Latent Space Network Models on Anisotropic Thurston Geometries

RESUMO

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.