Testing properties of trees in graphical models with covariance queries

2026-05-15

AI Summary

This work addresses the problem of efficiently verifying global structural properties of high-dimensional graphical models whose underlying graphs are trees, without requiring full reconstruction. Under the covariance query model, the authors propose the first randomized testing method that integrates statistical hypothesis testing with tree-structured graph-theoretic analysis. This approach achieves subquadratic query complexity for determining key properties such as the number of leaves, maximum degree, typical distance, and diameter. Notably, this study establishes the first explicit upper bounds on query complexity for a range of fundamental tree properties within the covariance query framework, substantially reducing the computational cost of structural testing in high-dimensional tree models.

Abstract

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.

Problem

• graphical models
• tree properties
• covariance queries
• structural testing
• query complexity

Innovation

• covariance queries
• graphical models
• tree testing
• sub-quadratic algorithms
• structural property testing