Dynamics on and of Complex Networks

Abstract

<p>Networks -- abstract objects composed of \emph{vertices} connected by \emph{edges}, are ubiquitous in the real world. </p><p>Examples such as social networks, the world wide web, and neural networks in the brain</p><p>are constantly evolving in their topology, the state of their vertices, or a combination of the two.</p><p>This dissertation presents a computational and theoretical study of three models of network dynamics, one corresponding to each of these modes of evolution.</p><p>The first study models the disintegration of a social network of voters with binary opinions, who prefer to be connected to others with the same opinion. </p><p>We study two versions of the model: the network evolves by voters in discordant ties choosing to either </p><p>adopt the opinion of their neighbors, or to rewire their ties to some randomly chosen voter of (i) the same, or (ii) any, opinion. </p><p>We examine how the probability of rewiring, and the initial fraction $\rho_{\textrm{i}}$ in the minority, </p><p>determine the final minority fraction $\rho_{\textrm{f}}$, when the network has bifurcated. </p><p>In case (i), there is a critical probability, that is independent of $\rho_{\textrm{i}}$, above which $\rho_{\textrm{f}}$ is unchanged from $\rho_{\textrm{i}}$, </p><p>and below which there is full concensus. </p><p>In case (ii), the behavior above the critical probability, that now depends on $\rho_{\textrm{i}}$, is similar; but </p><p>below it, $\rho_{\textrm{f}}$ matches the result of starting with $\rho_{\textrm{i}} = 1/2$. Using simulations and approximate calculations, we explain why these two nearly identical </p><p>models have such dramatically different behaviors.</p><p>The second model, called the \emph{quadratic contact process} (QCP) involves ``birth'' and ``death'' events on a static network. </p><p>Vertices take on the binary states occupied(1) or vacant(0). </p><p>We consider two versions of the model -- Vertex QCP, and Edge QCP, corresponding to </p><p>birth events $1-0-1 \longrightarrow 1-1-1$ and $1-1-0 \longrightarrow</p><p>1-1-1$ respectively, where `$-$' represents an edge. </p><p>We study the fraction of occupied vertices at steady state as a function of the birth rate, keeping </p><p>the death rate constant. To investigate the effects </p><p>of network topology, we study the QCP on homogeneous networks with a bounded or rapidly decaying degree distribution, </p><p>and those with a heavy tailed degree distribution. </p><p>From our simulation results and mean field calculations, we conclude </p><p>that on the homogeneous networks, there is a discontinuous phase transition with a</p><p>region of bistability, whereas on the heavy tailed networks, the</p><p>transition is continuous. Furthermore, the critical birth rate is positive </p><p>in the former but zero in the latter.</p><p>In the third study, we propose a general scheme for spatial networks evolving in order to reduce their total edge lengths. </p><p>We study the properties of the </p><p>equilbria of two networks from this class, one of which interpolate between two well studied objects: the Erd\H{o}s-R\'{e}nyi random graph, and the random geometric graph. </p><p>The first of our two evolutions can be used as a model for a social network where individuals have fixed opinions about a number of issues and adjust their ties to be connected to people with similar views. </p><p>The second evolution which preserves the connectivity of the network has potential applications in the design of transportation networks and other distribution systems.</p>Dissertatio