Model-oriented Graph Distances via Partially Ordered Sets

TL;DR

This paper introduces a new model-oriented distance for graphs based on partial orderings, providing a consistent metric that considers the model structure, applicable across various graph classes, and scalable algorithms for computation.

Contribution

It proposes a novel distance framework for graphs as statistical models using partial orders, improving over existing methods by capturing model structure and enabling scalable algorithms.

Findings

The proposed distance is a valid metric on graph spaces.

It outperforms existing distances in theoretical and empirical evaluations.

Algorithms for computing and bounding the distance are scalable to moderate-sized graphs.

Abstract

A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for combinatorial parameters such as graphs that represent statistical models. Defined on the graphs alone, existing proposals like the structural Hamming distance ignore the structure of the model space and can thus exhibit undesirable behaviors. We propose a model-oriented framework for defining the distance between graphs that is applicable across different graph classes. Our approach treats each graph as a statistical model and organizes the graphs in a partially ordered set based on model inclusion. This induces a neighborhood structure, from which we define the model-oriented distance as the length of a shortest path through neighbors, yielding a metric in the space of graphs. We apply this framework to probabilistic undirected graphs, causal directed acyclic graphs, probabilistic completed partially directed acyclic graphs, and causal maximally oriented partially directed acyclic graphs. We analyze theoretical and empirical behaviors of the model-oriented distance and draw comparison with existing distances. By exploiting the underlying poset structures, we develop algorithms for computing and bounding the proposed distance that scale to moderate-sized graphs.