Geometry & Geography of Complex Networks

Complex systems are made up of many interacting components. Network science provides the tools to analyze and understand these interactions. Community detection is a key technique in network science for uncovering the structures that shape the behavior of these networks. This thesis introduces the Adaptive Cut, a novel method that improves clustering methods by employing multi-level cuts in hierarchical dendrograms. Overcoming the limitations of traditional single-level cuts-especially in unbalanced dendrograms-the Adaptive Cut provides a multi-level cut by optimizing a Markov chain Monte Carlo with simulated annealing. In addition, we propose the Balanceness score, an information-theoretic metric that quantifies dendrogram balance and predicts the benefits of multilevel cuts. Evaluations on over 200 real and synthetic networks show significant improvements in partition density and modularity.
In the second part, we explore network geometry using machine learning techniques like network embeddings and graph neural networks. Focusing on the Danish cohabitation network, we use methods such as node2vec and Infomap to study community structures and the interplay between higher-dimensional network geometry and geography. Our analysis shows that incorporating network geometry allows redefining administrative boundaries into non-contiguous regions that better reflect social and spatial dynamics. We also discuss the representation of hierarchical data in hyperbolic space through Poincaré maps, which can represent tree-like structures in low dimension.
In addition, we examine how geography constrains human mobility, an aspect often overlooked in scale-free characterizations of mobility. By incorporating geography via the pair distribution function from condensed matter physics, we separate geographic constraints from mobility choices. Analyzing datasets containing millions of individual movements, we identify a universal power law that spans five orders of magnitude, thereby bridging the divide between distance-based and opportunity-driven models of human mobility.