Improved methods to compare distance metrics using uniform random spanning trees and their applications in network analysis

Abstract

This dissertation considers the network analytics problem of comparing two distance metrics on the same set of $n$ entities. The classical solution proposed in 1967 is the Mantel test: a permutation-based hypothesis test whose complexity of $\Theta(n^2)$ per permutation makes it intractable for even medium-sized data sets. Our first contribution is the development of a faster ($\Theta(n)$ per permutation) permutation test based on uniform random spanning trees called DIMECOST\textsubscript{P}. In addition, we present DIMECOST\textsubscript{CC} a method that once again uses uniform random spanning trees to compute correlation coefficients that quantify the strength of the linear relationships between distance metrics. The spanning tree based method eliminates dependencies in the data resulting from the triangle inequality making the correlation coefficient a more robust metric. We next develop mathematical formulas for special cases of transportation networks (line, circle and mesh graphs) based on the DIMECOST\textsubscript{CC} methodology. These are used to analyze trends in graph size and topology and location and amount of congestion in the network changes. Finally, we apply DIMECOST\textsubscript{CC} to assess fairness in the Denver area transportation network. Along with average speed to evaluate network effectiveness, we show that this approach can be used by transportation planners and policymakers to perform a quick screening of potential future projects in order to recommend a short list of projects for subsequent, extensive evaluations.