We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

翻译：我们考虑高维图模型底层图属性的测试问题。采用Lugosi、Truszkowski、Velona和Zwiernik（2021）提出的协方差查询模型，研究底层图为树的情况。本文主要结果表明，尽管重建整棵树可能成本高昂，但某些全局结构属性可以高效测试。具体而言，我们设计了针对全局结构属性的随机化测试，其查询次数低于二次复杂度。我们为若干基本属性开发了测试程序，包括树的叶子节点数、最大度、典型距离和直径。针对每个属性，我们获得了显式的查询复杂度界限，该界限依赖于目标阈值和容差参数。