Geometry and 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, 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.

💡 Research Summary

This dissertation tackles two intertwined themes in network science: community detection and the geometric‑geographic analysis of complex networks. The first part reviews fundamental concepts (adjacency matrices, degree distributions, clustering coefficients) and surveys classic community detection methods such as modularity‑based Louvain, information‑flow Infomap, stochastic block models, and link‑clustering. Building on this foundation, the author introduces the Adaptive Cut algorithm, a multi‑level cutting strategy for hierarchical dendrograms. Instead of a single cut, Adaptive Cut explores many possible cuts across different levels. It frames the selection problem as a Markov‑chain Monte Carlo (MCMC) process combined with simulated annealing: candidate cuts are proposed, accepted probabilistically, and the temperature schedule gradually shifts from exploration to exploitation. The quality of a cut is measured by partition density for link‑clustering and modularity for node‑based clustering.

To predict when multi‑level cuts will be beneficial, the dissertation proposes the Balanceness score, an information‑theoretic metric that quantifies how evenly a dendrogram splits its nodes. High Balanceness indicates that a single‑level cut would likely miss important sub‑communities, making a multi‑level approach advantageous. Empirical evaluation on more than 200 real‑world and synthetic networks—including social, infrastructural, biological, Erdős‑Rényi, and scale‑free models—shows that Adaptive Cut consistently improves modularity and partition density by 5‑12 % over baseline methods. The gains are especially pronounced for highly unbalanced dendrograms, where traditional cuts either over‑merge or over‑split communities.

The second part shifts focus to network geometry. It surveys low‑dimensional embedding techniques: matrix factorization, Laplacian eigenmaps, node2vec, and latent distance models. The author emphasizes hyperbolic embeddings (Poincaré disk) because hyperbolic space can represent tree‑like hierarchies with low distortion. Using the Danish co‑habitation network as a case study, the author demonstrates that embedding‑based clustering can redefine administrative boundaries into non‑contiguous regions that better capture underlying social and spatial affinities. This suggests that geometric representations can inform policy‑relevant territorial redesign beyond purely geographic constraints.

The third part addresses human mobility. By borrowing the pair distribution function (PDF) from condensed‑matter physics, the author separates pure geographic distance constraints from choice‑driven opportunity effects. Analyzing millions of individual movement records, the PDF reveals a universal power‑law distance distribution spanning five orders of magnitude with an exponent around 1.6. This finding bridges the gap between distance‑based gravity models and intervening‑opportunity models, offering a unified statistical description of mobility that accounts for both spatial friction and opportunity selection. The PDF framework also enables quantitative assessment of how city size, transport infrastructure, and population density modulate mobility patterns.

In conclusion, the dissertation makes three major contributions: (1) Adaptive Cut and the Balanceness metric, which together raise the accuracy and interpretability of community detection, especially in networks with multi‑scale structure; (2) a demonstration that hyperbolic and other low‑dimensional embeddings can uncover latent geometric organization, supporting the redesign of administrative units and enhancing downstream tasks such as link prediction and node classification; (3) a novel, physics‑inspired approach to human mobility that reconciles competing theoretical models and uncovers a robust scaling law across diverse contexts. Limitations include the computational cost of MCMC sampling for very large graphs and the need for more rigorous validation of hyperbolic embeddings’ interpretability. Future work should explore scalable parallel implementations of Adaptive Cut, develop quantitative metrics for hyperbolic embedding quality, and integrate socioeconomic variables into the mobility PDF framework.