Manifold learning for coarse-graining networks and for parameter reduction

Abstract

Recent decades have seen a tremendous rise in the affordability and performance of various computational technologies, enabling researchers to propose and probe ever more complicated numerical models. These simulations often generate incredible quantities of data that must be sifted through to glean useful conclusions. This thesis highlights our efforts to automate this process in two specific areas: (a) uncovering simplified descriptions of dynamic network models and (b) detecting important parameter combinations in general nonlinear systems. Both advances involve modification of the manifold learning algorithm, Diffusion Maps (DMAPS), to address the particular problem.

In the first case, the challenge is quantifying the similarity of two networks in a reasonable amount of computational time. We propose a number of possible solutions, and examine their performance when combined with DMAPS. We find that by combining suitable measures of similarity with DMAPS we are able to uncover low-dimensional structure in a set of networks, thus enabling us to describe the system in terms of one or two values instead of thousands.

In the second, we must extend Diffusion Maps to operate on the graph of a function. In particular we consider a model that maps parameter values to some output. By properly formulating the DMAPS kernel, we enable DMAPS to discover the directions in parameter space along which model predictions vary most significantly.

Both result in algorithms that we hope are practically useful to researchers in a variety of fields who are looking for simplified descriptions of their complex systems.