Topology Design for Optimal Network Coherence

We consider a network topology design problem in which an initial undirected graph underlying the network is given and the objective is to select a set of edges to add to the graph to optimize the coherence of the resulting network. We show that network coherence is a submodular function of the network topology. As a consequence, a simple greedy algorithm is guaranteed to produce near optimal edge set selections. We also show that fast rank one updates of the Laplacian pseudoinverse using generalizations of the Sherman-Morrison formula and an accelerated variant of the greedy algorithm can speed up the algorithm by several orders of magnitude in practice. These allow our algorithms to scale to network sizes far beyond those that can be handled by convex relaxation heuristics

On the Smallest Eigenvalue of Grounded Laplacian Matrices

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $\Theta(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes

A network model of mass media opinion dynamics

The coexistence of diverse opinions is necessary for a pluralistic society in which people can confront ideas and make informed choices. The media functions as a primary source of information, and diversity across news sources in the media forms the basis for wider discourse in the public. However, due to numerous economic and social pressures, news sources frequently co-orient their content through what is known as intermedia agenda-setting. Past research on the subject has examined relationships between individual news sources. However, to understand emergent behaviour such as opinion diversity, we cannot simply analyse individual relationships in isolation, but instead need to view the media as a complex system of many interacting entities. The aim of this thesis is to develop and empirically test a method for understanding the network effects that intermedia agenda-setting has on the diversity of expressed opinions within the media. Utilising latent signals extracted from news articles, we put forward a methodology for inferring networks that capture how agendas propagate between news sources via the opinions they express on various topics. By applying this approach to a large dataset of news articles published by globally and locally prominent news organisations, we identify how the structure of intermedia networks is indicative of the level of opinion diversity across various topics. We then develop a theoretical model of opinion dynamics in noisy domains that is motivated by the empirical observations of intermedia agenda formation. From this, we derive a general analytical expression for opinion diversity that holds for any network and depends on the network's topology through its spectral properties alone. Finally, we validate the analytical expression in a linear model against empirical data. This thesis aids our understanding of how to model emergent behaviour of the media and promote diversity