发表机构
Yale University; The Wharton School, University of Pennsylvania（耶鲁大学; 宾夕法尼亚大学沃顿商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非乘积非等曲率三维几何中潜在空间网络模型的可辨识性，提出锚点消除等距模糊，证明有向连边可一阶检测几何，耦合模型优于排序模型。
AI 中文摘要
潜在空间网络模型将节点置于度量空间中，并让连边概率随距离增加而减小。在常曲率空间中，成对距离决定位置（相差一个等距变换）。在乘积空间以及其余三种三维模型几何中，情况并非如此。我们研究既非常曲率也非乘积的三种几何：Heisenberg群、可解群以及双曲平面单位切丛的万有覆盖。我们询问一个网络能识别关于其中位置的什么信息，以及其几何何时可被检测。两个锚点消除了等距模糊性。小配置不由其距离决定，而超过有限阈值后，一般局部可辨识性成立，并在Heisenberg群中得到验证。我们推导出等距群商空间上的后验分布。对于小配置，到最近乘积或常曲率竞争者的散度是曲率差别的应力分量，并在尺度的高阶上消失；因此，无向连边仅在大网络且具有大封闭面积时才检测到几何。有向连边在一阶上暴露几何：在两种扭曲几何中，非对称偏好围绕三角形循环，与封闭面积成正比，这是任何加性排序都无法产生的。在稠密竞争性博弈反例网络上，耦合模型在每种几何上都优于排序、度修正排序和自由反对称项。具有相同耦合项的欧几里得模型与其匹配，因此增益在于通过共享坐标耦合相似性和循环性。一个加性和乘性效应模型预测得更好。
英文摘要
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.
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