Spatial Random Graphs: Studying dependence structures in spatial data
by Leoni Carla Wirth
Date of Examination:2025-09-11
Date of issue:2025-10-02
Advisor:Prof. Dr. Dominic Schuhmacher
Referee:Prof. Dr. Dominic Schuhmacher
Referee:Prof. Dr. Gesine Reinert
Persistent Address:
http://dx.doi.org/10.53846/goediss-11528
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Abstract
English
Spatial random graphs provide a powerful framework for analyzing the relations and interactions governed by an underlying spatial structure. Such graphs can be used not only to model and analyze observed networks, but also to represent complex data via synthetic networks, i.e. networks in which the edge structure is not observed but in which edges are introduced to provide a representation of additional information. In both cases, spatial graph models can be very useful for describing and highlighting the characteristics of the underlying dependence structures. A wide range of sophisticated spatial graph models is important for capturing some key structures of complicated real-world networks. This thesis makes substantial contributions to the theoretical study of dependence structures in spatial (random) graphs. It is based on three research articles (currently available as preprints on arXiv) and one work in progress. The thesis summarizes the approaches and main results of these papers, which are included in full in the appendix. A fundamental ingredient for understanding and analyzing the graph structure is a metric that measures the (dis)similarity between spatial random graphs in a natural way. The first research article constructs such a spatial graph metric by building on an optimal assignment of the vertices with costs based on the dissimilarity of both, vertex and edge structures. The paper demonstrates that the new graph metric can be used as a statistical tool to infer important graph properties and further serves as a foundation for deriving a metric for spatial random graph distributions. Due to the complexity of the underlying dependence structure, a direct analysis of spatial random graphs is often not feasible. Instead, the second paper studies the approximation of possibly more complicated spatial random graphs with simpler spatial random graphs. These simpler spatial random graphs are based on suitable independence assumptions and are a generalization of the random geometric graph consisting of vertices given by a Gibbs process and (conditionally) independent edges generated from a connection function. The results are derived by an application of Stein's method which allows to state explicit convergence rates with respect to the metric for spatial random graph distributions introduced in the first paper. An application of these results yields interesting convergence statements, for example, in the context of thinnings and discretizations of spatial random graphs. Another application is considered in the third paper, where synthetic networks are studied under a privacy constraint. In this context, a private synthetic graph generator is introduced that takes observed attributes as an input and jointly generates a network based on the true data and private data, respectively. Finally, as part of ongoing research, it is outlined how kernel Stein discrepancies can be used to perform goodness-of-fit tests for point processes.
Keywords: Spatial random graphs; Spatial graph distance; Stein's method; Spatial random graph approximation; Differential privacy; Synthetic network generation; Goodness-of-fit testing
