Abstract

Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied in biological contexts in recent years.

In network science, most tools focus solely on local properties based on pairwise connections, the topological tools reveal more global features. I apply persistent homology to biological networks to see which properties these tools can uncover, which might be invisible to existing methods. In my talk I will show the use of three different methods from Computational Topology, so called filtrations: a filtration by weights, a weight rank clique filtration and a Vietoris-Rips filtration to analyse networks. The example networks I apply these tools to consist of fMRI data from neuroscientific experiments on human motor-learning and the study of schizophrenia, a mathematical oscillator model (the Kuramoto model), as well as imaging data from tumour blood vessel networks. In all cases I will show how these tools reveal insights into the biology or dynamics of the studied problems.