On the Properties of Gromov Matrices and their Applications in Network Inference

TL;DR

This paper introduces Gromov matrices to represent weighted trees, enabling a new, algebraic approach to improve network inference by considering multiple spanning trees simultaneously, enhancing accuracy while maintaining computational efficiency.

Contribution

It proposes a novel method using convex combinations of Gromov matrices to improve network inference over traditional spanning tree heuristics.

Findings

Enhanced inference accuracy by considering multiple spanning trees.

Method remains computationally tractable due to simple algebraic matrix operations.

Demonstrated usefulness in network inference and estimation applications.

Abstract

The spanning tree heuristic is a commonly adopted procedure in network inference and estimation. It allows one to generalize an inference method developed for trees, which is usually based on a statistically rigorous approach, to a heuristic procedure for general graphs by (usually randomly) choosing a spanning tree in the graph to apply the approach developed for trees. However, there are an intractable number of spanning trees in a dense graph. In this paper, we represent a weighted tree with a matrix, which we call a Gromov matrix. We propose a method that constructs a family of Gromov matrices using convex combinations, which can be used for inference and estimation instead of a randomly selected spanning tree. This procedure increases the size of the candidate set and hence enhances the performance of the classical spanning tree heuristic. On the other hand, our new scheme is based on simple algebraic constructions using matrices, and hence is still computationally tractable. We discuss some applications on network inference and estimation to demonstrate the usefulness of the proposed method.