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

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. Index TermsGromov matrix, spanning tree, network inference and estimation

Identifying misinformation and their sources in social networks

“…More discussions on the properties of Gromov matrix can be found in [109]. For each node in V \V we need to find two of their BFS trees, which can be com- Find two BFS trees rooted at s in opposite search directions to obtain Λ 1 s and Λ 2 s .…”

“…Fromthe Corollary 1 of[109], we see that the convex combination of two Gromov matrices may not be a Gromov matrix, unless one of them is diagonal. Therefore M s is in general not a Gromov matrix.…”

“…Therefore M s is in general not a Gromov matrix. However,[109] provided a procedure called G-convex to modify some entries of M s to obtain a Gromov matrix. If the modified M s is used, then A s is a Gromov matrix of a weighted tree (though may not be a subtree of G) since H s is diagonal.…”

“…If the modified M s is used, then A s is a Gromov matrix of a weighted tree (though may not be a subtree of G) since H s is diagonal. Through a case study, the paper[109] provided further insights into how well convex combinations of Gromov matrices can approximate an information propagation path in a network. We refer the interested reader to paper[109] for more details.…”

“…Through a case study, the paper[109] provided further insights into how well convex combinations of Gromov matrices can approximate an information propagation path in a network. We refer the interested reader to paper[109] for more details. In our thesis, we do not apply G-convex so that A s may not be a Gromov matrix.…”