Geometry and Geography of Complex Networks
Louis Boucherie
TL;DR
This work investigates the intersection of network structure, geometry, and geography. It introduces the Adaptive Cut to perform multilevel dendrogram cuts via MCMC with simulated annealing and the Balanceness score to assess dendrogram balance, improving partition density and modularity across many networks. The second part leverages network embeddings, including hyperbolic representations and Poincaré maps, to reveal that geography can redefine administrative boundaries and to analyze mobility through a geography-aware pair distribution framework, uncovering a universal five-order-of-magnitude power law that bridges distance-based and opportunity-driven mobility models. Overall, the thesis demonstrates that incorporating geometry and geography into network analysis yields more faithful representations of social-spatial dynamics and scalable, interpretable models for large-scale mobility data.
Abstract
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, 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 Poincare 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.