Forman-Ricci Curvature and Associated Geometric Flows for Complex Networks


With the rise of data science in the last decades, networks have emerged as a powerful tool to represent and study complex systems. This thesis presents geometric tools for characterizing complex networks through the analysis of so far widely neglected network properties to provide novel insights into the structure and evolution of complex networks. Firstly,we introduce Forman-Ricci curvature and its corresponding geometric flows as a characteristic for complex networks, thus extending the common approach of node-based network analysis with edge-based characteristics. Following a theoretical introduction and mathematical motivation, we apply the proposed network-analytic methods to static and dynamic complex networks and compare the results with established node-based characteristics. Our work suggests a number of applications for data mining including denoising and clustering of experimental data, as well as extrapolation of network evolution. In the second part, we extend the introduced formalism to higher dimensions in an attempt to study the shape of networks. While there exists a body of knowledge concerned with local network characteristics, the systematic study of global network structure is a largely untouched field. Here, we introduce theoretical tools to classify the shape of a given network based on Forman’s Ricci curvature for graphs and its associated Ricci flow. Our setting allows for a network-theoretic formulation of a Gauß-Bonnet style theorem and the computation of Euler characteristics. Attempting to describe the long-term behavior of networks qualitatively, we introduce a ‘gauge theory’ for networks that gives rise to a global classification scheme based on Ricci curvature. Besides highlighting theoretical aspects, we discuss important implications for the study of real-world networks with an emphasis on correlation networks as used in the biological sciences and social network analysis.

Undergraduate Thesis, U Leipzig.