Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Lindi M. Wahl

Abstract

Complex network theory offers useful approaches to analyze the structural and functional properties of real life networks. In this work, we explore some of the mathematical concepts of network theory and study real life systems from a complex network perspective. We pay particular attention to networks of connections within the human brain. We analyze weighted networks calculated from full functional magnetic resonance imaging (fMRI) data acquired during task performance. The first novelty of this study is the fact that we retain all of the connections between all of the voxels in the full brain fMRI data. We then evaluate the extent to which this rich dataset can be described by existing models of scale-free or exponentially truncated scale-free topology, comparing results across a large number of more complex topological models as well. Our results suggest that the novel model presented in this dissertation offers a significantly better fit at the voxel level.

Structural characterization of the brain can also give insights into the effects of traumatic injuries. For our second study, we used brain networks to explore the topological consequences of brain damage in computational models. By simulating the effects of several traumatic brain injury (TBI) models we find a variety of disruptions in the modular structure and connectivity of the post-injury networks. In particular, we focused on the effects of focalized injuries, axonal degeneration and diffuse microlesions. Our results suggest that mathematical models can predict, to some extent, the structural and functional effects of TBIs based on the analysis of specific topological measures. Furthermore, these results may be correlated to known cognitive sequelae of TBI. In order to complement the set of topological properties studied, we review the concept of participation of a node in a modular network. The participation coefficient is often used to provide a ranking of the "importance" of a node; however, we propose a new measure, called the gateway coefficient, to assess the involvement of a node. Our results suggest that the gateway coefficient has a superior ranking power than the participation coefficient since it takes into account a variety of modular properties of the network. These results are illustrated with examples of simulations and real life datasets, such as the air transportation network and the human brain network.

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