Topological tools for understanding complex systems

The behavior of complex systems is often influenced by their structure. In mathematics, the field of algebraic topology has been especially useful for characterizing mathematical structures. Topological data analysis (TDA) is a growing field in which methods from algebraic topology are applied to studying the structure of data. TDA has been used in a variety of applications, including biological data, granular materials, and demography. Social interactions are heavily informed by space and have complex structure due to patterns in the way humans arrange themselves geographically. Consequently, social applications can benefit from the application of TDA.In this dissertation, I develop topological methods for studying spatial networks and apply them to a wide variety of data sets. In particular, I study methods for building topological spaces (specifically, simplicial complexes) based on data. I present two novel simplicial-complex constructions, the adjacency complex and the level-set complex, for spatial data. I apply both constructions to random networks, cities, voting, and scientific images, gaining insights into the structure of these systems. I also propose a novel simplicial complex construction for studying patterns of neighborhood formation based on combining demographic and spatial data. I present case studies in neighborhood segregation for two U.S. cities. In addition to my topological research, I discuss two projects in the study of social systems using methods from network analysis. I present an extension to multilayer networks of the Hegselmann--Krause model for opinion dynamics and discuss preliminary findings on its convergence properties. I also present a framework for estimating homelessness underreporting in California Local Education agencies (LEAs)

Clustering in a network of non-identical and mutually interacting agents

Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behaviour. It is observed in fields ranging from the exact sciences to social and life sciences; consider, for example, swarm behaviour of animals or social insects, the dynamics of opinion formation or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modelling brain or heart cells. We consider a clustering model with a general network structure and saturating interaction functions. We derive both necessary and sufficient conditions for clustering behaviour of the model and we investigate the cluster structure for varying coupling strength. Generically, each cluster asymptotically reaches a (relative) equilibrium state. We discuss the relationship of the model to swarming, and we explain how the model equations naturally arise in a system of interconnected water basins. We also indicate how the model applies to opinion formation dynamics