Geometry-Inspired Measures for Network Science

Abstract

Complex networks provide a useful framework to study a wide range of social, biological and technological systems. Almost two decades ago, network science has emerged as a new interdisciplinary field with the goal to unravel universal properties of complex networks. The view of networks as finite metric spaces opens several opportunities towards applying tools from geometric analysis to study the structure of networks. Ricci curvature is an essential quantity in Riemannian geometry which encodes all the key properties of the underlying metric space. Recently, a few discretizations of the classical Ricci curvature have been introduced in the context of complex networks. The main aim of this work is to discover the applications of discrete Ricci curvatures in real-world networks. For this purpose, we analyse networks of global financial indices and networks of functional connectivity in the human brain. We study the temporal evolution of discrete Ricci curvatures in networks of global financial indices and show that these measures can effectively be used as indicators of crashes and bubbles in the global financial market. We also study the changes in discrete Ricci curvatures of functional connectivity networks in brains with autism spectrum disorder and ageing brains. We find that discrete Ricci curvatures can be used to determine abnormal connectivity patterns in the human brain. We also study other widely used network measures to gain more insights about the topological properties of networks of global financial indices and networks of functional connectivity in the human brain.