Limit theory for spatial random networks

Abstract

Samenvatting viAs the complexity and size of networks relevant for TDA increases, the need for a better comprehension of their behavior on a large scale becomes more and more important. Spatial networks can be modeled by arranging nodes randomly in a metric space and establishing connections based on spatial constraints or probabilities. The generated object is referred to as a spatial random network. This thesis aims to contribute to the understanding of spatial random networks by studying their limit theory. A major focus of this thesis is the analysis of rare events. Relatively small deviations from what is expected are the territory of central limit theorems. Rarer events can be approached with Poisson approximations or the theory of large deviations. The latter is the main focus of this thesis. vii Large deviationsThe theory of large deviations is a branch of probability theory that aims to quantify the probability of rare events. It provides powerful tools to study the asymptotic behavior of stochastic processes and has applications in a wide range xi More details about Poisson approximations can also be found in [8]. A first use case of the Poisson approximation in the context of rare events analysis goes back to [18], studying, among other things, the number of soldiers in the Prussian army killed by horse kicks.The Poisson approximation can be extended to a Poisson process approximation as outlined for instance in [23]. A Poisson process approximation in this spirit is derived in Chapter 2. For this, we need a notion of distance between two point xiii