Real equiangular lines and related codes

We consider real equiangular lines and related codes. The driving question is to find the maximum number of equiangular lines in a given dimension. In the real case, this is controlled by combinatorial phenomena, and until only very recently, the exact number has been unknown. The complex case appears to be driven by other phenomena, and configurations are conjectured always to meet the absolute bound of d^2 lines in dimension d. We consider a variety of the techniques that have been used to approach the problem, both for constructing large sets of equiangular lines, and for finding tighter upper bounds. Many of the best-known upper bounds for codes are instances of a general linear programming bound, which we discuss in detail. At various points throughout the thesis, we note applications in quantum information theory

Network effects in mass communication - an analysis of information diffusion in markets

In this thesis we investigate the diffusion of information like news, announcements, and commercials in social networks. Such information propagates through a mix of mass communication and interpersonal communication. For example, people who watch a TV spot about a new car will discuss it with their friends. Both communication methods influence the awareness, preferences, and opinions that people display towards certain topics, products, and services. The effects of mass and inter-personal communication on the diffusion process have been studied intensively in several areas, for example, in sociology, economics, social psychology, political science, and marketing. Most of these studies highlight the role of inter-personal relation structures, that is, the network of social ties, in the diffusion process. However, a concise diffusion model that quantifies the effects of social networks and helps to improve mass communication towards structured populations is still in demand. Our purpose is first to analyse the drivers of social networks, then to model the diffusion of information on social networks, and finally to quantify the network effects on the diffusion process. We describe and construct social networks as graphs and present anthropological, psychological, and random factors that shape them. Based on one of these factors, structural balancing, we propose an evolutionary model of social networks, suggesting that the structure of social networks can change dramatically over time. For modelling diffusion processes on social networks, we follow a two-step procedure. We first combine three different generation methods, the generalised random graph, the small-world model, and a third method (random graph with a given assortment structure) to design realistic networks. Then we simulate the propagation of information on these networks. As the computer requirements for such simulations can be expensive, we introduce an efficient computer algorithm that is widely applicable to complex diffusion studies in markets, organisations, and societies. One result of the simulations is a robust closed-form approximation to the diffusion's trajectory in networks. Such an approximation allows marketing and PR managers to predict aggregate market outcomes such as the popularity of a commercial through surveys prior to the launch of a promotional campaign. The simulations also indicate the impact of the network's structure on the diffusion. To measure the network effects on the propagation of information, we run regression analyses with the communication intensity and the different network features as explanatory variables. These network features are the degree distribution, the transitivity (clustering), degree correlation, and the average path length. The regressions show, above all, that network effects are conditional on the intensity of mass communication: the less intensive mass communication, the more important become network effects. For mass communication typical in marketing and PR, the network structure can have a strong impact on the diffusion process. The regressions quantify the respective contribution of each network feature on the diffusion process over time. Our findings confirm and partly reconcile contradictionary results of comparable studies in epidemics and sociology. Finally, our analysis allows us to prioritise different network effects. This can be useful in various situations, for example, when estimating a diffusion process with incomplete network data