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Title:  Dynamical systems on networks 
Author(s):  Ferguson, Timothy 
Director of Research:  Bronski, Jared 
Doctoral Committee Chair(s):  DeVille, Lee 
Doctoral Committee Member(s):  Zharnitsky, Vadim; Laugesen, Richard 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Dynamics
Networks 
Abstract:  We obtain results for two distinct dynamical models: the Kuramoto model, a general model for coupled oscillator systems, and a model for opinion formation in social networks. Our main focus is on understanding the fixed points of these systems and their stability. For many models the stability of such fixed points can be studied with a Laplacian matrix. We give a formula for the inertia of these matrices, characterizing the real parts of the spectrum, by relating them to another matrix depending on the network topology. We then study the Kuramoto model, and in particular, the phenomena of synchronization, when all oscillators rotate at a common frequency, which corresponds to a fixed point. This phenomenon is wellknown to depend on the natural frequencies of the oscillators and, more specifically, that the chance of synchronization increases if the natural frequencies are more similar. We then give upper and lower bounds for the volume of the set such frequencies in frequency space. Our bounds can be formulated in terms of sums over spanning trees which we further use to deduce that the volume is intimately related to the number of spanning trees for dense networks. We also characterize the structure of fixed points of the Kuramoto model by showing that every fixed point corresponds to a lattice point in a certain set which records how the phaseangles wrap around cycles in the network. As a consequence, under mild conditions, we derive the rate of growth of the number of fixed points as we consider increasingly large graphs with fixed topology. We also consider a model for opinion formation in social networks. More specifically, we characterize the global minima of an energy functional, intuitively the ``most stable" configurations, when the network is ``balanced" as well as show that the number of stable configurations can increase as we increase the strengths of the relationships in the network. Finally, we describe an algorithm for generating certain random networks. These networks are generalizations of Erd\H{o}sR\'{e}nyi graphs with correlations between pairs of edges depending on the particular pattern they create. We then use this algorithm to study the effect on fixed points of network properties and therefore the dynamics of the Kuramoto model. 
Issue Date:  20180413 
Type:  Text 
URI:  http://hdl.handle.net/2142/101153 
Rights Information:  Copyright 2018 Timothy Ferguson 
Date Available in IDEALS:  20180904 20200905 
Date Deposited:  201805 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois