| dc.description.abstract | Graphs are mathematical structures comprised of a set of nodes connected by edges, and
network science is the application of graph theory to real world data. Networks are used
as a model to analyse how entities, either individual actors, or complex systems, interact
with one another.
The research here will consist of extracting networks (we will use the terms \graph" and
\network" interchangeably) from ordered series, which we will focus on series ordered
by time. We will either do this with the aid of the visibility graph, which is a method,
based on visibility, for mapping a time series in to a graph, or through estimating the
wavelet correlation, a more conventional method used in neuroscience. The aim is to
describe the structure of time series and their underlying dynamical properties in graphtheoretical
terms, and then using this motivation to analyse large data sets spanning
several disciplines. We will describe a method, using the visibility graph, for quantifying
reversibility of non-stationary processes and apply this method to a large nancial data
set, with the intent of ranking companies based on their irreversibility. We also use the
visibility graph to develop a method which e ciently quanti es the asymmetries between
minima and maxima in time series, and we then apply the method to a variety of data
sets. Continuing with the theme of visibility, we study the spectral properties of visibility
graphs extracted from trajectories of the logistic map undergoing a period-doubling route
to chaos (known as the Feigenbaum scenario). Finally, we will use wavelet correlation to
construct networks from fMRI time series, and examine community structure with the
aim of di erentiating between brain networks of patients with schizophrenia from control
subjects. The general format throughout this thesis will start with theory, followed by
extensive numerical simulations, which we can then apply the methods to real data sets. | en_US |
