CHARACTERIZING THE SPECTRAL RADIUS OF A SEQUENCE OF ADJACENCY MATRICES BY WILLIAM D. FRIES A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Mathematics and Statistics May 2018 Winston-Salem, North Carolina Approved By: Miaohua Jiang, Ph.D., Advisor Kenneth Berenhaut, Ph.D., Chair Grey Ballard, Ph.D. Acknowledgments I would like to thank Dr. Miaohua Jiang for his guidance and expertise through- out the research experience. This project would not have been possible without him. I would also like to thank Dr. Kenneth Berenhaut, Dr. Grey Ballard, Dr. John Gem- mer, Dr. Sarah Raynor and the rest of the Mathematics and Statistics department at Wake Forest University for helping me realize the endless opportunities that studying math can bring. This work is dedicated to my parents, Karen and Andy, my sister, Margaret, and my brother, Jack: thank you for teaching me that some of best things in life come from its unpredictable nature. ii Table of Contents Acknowledgments . ii Abstract . iv List of Tables . v List of Figures . vi Chapter 1 Introduction . 1 1.1 Foundations of Network Epidemics . .2 1.1.1 Continuous Compartmental Epidemic Models . .2 1.1.2 Epidemiology on Networks . .5 1.1.3 Existing Bounds on the Largest Eigenvalue . 10 1.2 Constructing our Problem . 11 1.2.1 Terms and Definitions . 11 1.2.2 Defining the Transformation . 13 Chapter 2 Main Results . 16 2.1 Motivation for the Problem . 16 2.2 Characterization of Eigenvalues . 18 2.2.1 The Characteristic Polynomial . 18 2.2.2 Properties of x(m)........................ 24 2.2.3 Special Cases . 30 Chapter 3 Further Results and Applications . 37 3.1 Corollaries . 37 3.2 Applications . 41 3.2.1 Relative Size of Eigenvalue . 42 Chapter 4 Conclusions . 46 4.1 Future Work . 46 Bibliography . 48 Curriculum Vitae . 50 iii Abstract In this paper we explore the introductory theory of modeling epidemics on networks and the significance of the spectral radius in their analysis. We look to establish properties of the spectral radius that would better inform how an epidemic might spread over such a network. We construct a specific transformation of networks that describe a transition from a star network to a path network. For the sequence of adjacency matrices that describe this transition, we show the spectral radius of these graphs can be given in a simple algebraic equation. Using this equation we show the spectral radius increases as the star unfolds and establish bounds on the spectral radius for each network. iv List of Tables 1.1 Common Compartmental Epidemic Models . .2 3.1 Numerical Approximations for d50;k, 27 < k < 47. 44 3.2 Numerical Approximations for d75;k, 52 < k < 72 . 44 3.3 Numerical approximation d100;k, 77 < k < 97. 45 v List of Figures 1.1 The differential equationx _ = :1x − :2x2 for 0 ≤ x ≤ 1 . .4 1.2 The solution curves tox _ = :1x − :2x2 with varying initial conditions .5 1.3 The differential equationx _ = −:05x − :2x2 for 0 ≤ x ≤ 1 . .6 1.4 The solution curves tox _ = :25x − :2x2 with varying initial conditions7 1.5 The network and associated adjacency matrix for A8(4) . 12 1.6 The network and associated adjacency matrix for B6 ......... 12 1.7 n-degree star and associated adjacency matrix . 13 1.8 The network and associated adjacency matrix for An(2) . 14 1.9 The network and associated adjacency matrix for An(3) . 14 1.10 The network and associated matrix after the i − 1 unfolding . 15 1.11 The network and associated matrix after n − 3 unfolding actions . 15 1.12 The network and associated matrix after n − 2 unfolding actions . 15 2.1 Plots of the ρ(An(k)) as k varies for n = 50, 75; and 100. 17 3.1 Numerical Approximations for ρ(A50(k)) . 42 3.2 Numerical Approximations for ρ(A75(k)) . 43 3.3 Numerical Approximations for ρ(A100(k)) . 43 3.4 A graph of n vs. maxk(pn(k)) ...................... 44 vi Chapter 1: Introduction Recent research into the impact of the how the spectral radius of a network on the spread of an epidemic in such network [1, 2, 3] motivates our research into the spectral radius of trees. If we consider a population of agents who are susceptible to an epidemic and whose connections are represented by the adjacency matrix A = [aij] where 8 β if node i is adjacent to node j <> ij aij = δi if i = j (1.1) :>0 else with βij being the probability that if agent i is infected that agent j becomes infected in one time-step (∆t) and δi by the probability that i recovers in ∆t, then there is a strong relationship between the largest eigenvalue of A and the reproduction number, the initial rate at which the epidemic spreads through the network [1, 2, 3, 4, 5, 6]. This prompts the question: how does the network structure affect the largest eigenvalue and ultimately how the disease will spread through the population? Re- search in graph theory has shown the maximal and minimal configurations of trees along with inequalities describing how spectral radii of related networks are related [7]. Our question restricts our networks to a specific set of trees, all with n vertices and can be described as stars each having one long arm [8]. We ask: how does the largest eigenvalue change as we transform our graph from a star to a path and can we find bounds for the largest eigenvalue? 1 1.1 Foundations of Network Epidemics 1.1.1 Continuous Compartmental Epidemic Models Before considering epidemics spreading across networks, it is useful to consider the case when a disease can be transmitted from anyone to anyone. These models are commonly referred to as fully-mixed, and the simplest epidemic models are the SI, SIR, SIS, and SIRS models (Table 1.1.1). `S' refers to the susceptible, `I' refers to infected, and `R' refers to recovered or removed. The order of the letters describe how a member of the population might move through different stages of a disease. Thus the SIRS model would model a disease in which someone might catch the disease, recover with a brief stage of immunity and then return to the susceptible population. In our models, we refer to s as the percent of the population that is susceptible, x, the percent of the population that is infected, and r as the percent of the population that is recovered or removed from the system. SI SIR s_ = −βsx s_ = −βsx x_ = βsx − δx x_ = βsx r_ = δx s_ = −βs(1 − s − x) x_ = β(1 − x)x r_ = δ(1 − s − x) SIS SIRS s_ = ηr − βsx s_ = δx − βsx x_ = βsx − δx x_ = βsx − δx r_ = δx − ηr x_ = (β − δ − βr − βx)x x_ = (β − δ)x − βx2 r_ = δx − ηr Table 1.1: The dynamical systems for simple epidemic models. These are commonly referred to as compartmental epidemic models because they 2 separate the population into compartments. This is to be contrasted with the agent- based epidemic model in which each agent's transmission and recovery rates are in- dependently determined which we will discuss later. The solution to the SI model is well known. Using separation of variables and initial condition x(0) = x0, the solution can be given as βt x0e x(t) = βt 1 − x0 + x0e The SI model has two clear fixed points, one unstable at x = 0 and one stable at x = 1 for any β > 0. This implies that, if the disease spreads, then eventually almost everyone will become infected. If we consider the SIS model, solutions can be given by: δ Ce(β−δ)t x(t) = 1 − (1.2) β 1 + Ce(β−δ)t with C = βx0 [1]. The bifurcation parameter to this equation, commonly referred β−δ−βx0 β to as the Reproduction Number, is given by R0 = δ , and the bifurcation occurs at R0 = 1 [1]. To apply this theory we consider the example below. Example 1. A small number, x0 essentially 0, are discovered to have a disease with a transmission rate of :2 and a recovery rate of :1 and can be modeled by the SIS model. Clearly the disease will spread and will follow the equation: 1 Ce:1t x(t) = 2 1 + Ce:1t The differential equation can be seen in Figure 1.1 and a sample of its solution curves can be seen in Figure 1.2. When analyzing this graph, we notice that we will have a fixed point at x∗ = :5 3 0.2 0.4 0.6 0.8 1.0 -0.02 -0.04 -0.06 -0.08 -0.10 Figure 1.1: The differential equationx _ = :1x − :2x2 for 0 ≤ x ≤ 1 which is a stable fixed point. That is, over time, about half of the population will be infected. Notice that if we change δ = :25 our differential equation becomes Figure 1.3 and our solution curves can be seen in Figure 1.4. Clearly, we have passed the bifurcation value and our epidemic will be eradicated. We also can see that as we approach the bifurcation value (R0 = 1), the stable fixed point at (β − δ) approaches 0. When comparing this to the previous example, we notice we have only one fixed point, at x∗ = 0. This implies the percent of the population that is infected tends β to 0 as time progresses.