# Revealing structure-function relationships in functional flow networks via persistent homology

## TL;DR

This paper uses persistent homology to analyze flow networks, revealing hidden topological features that relate network structure to function, and providing universal insights into multifunctionality across diverse biological networks.

## Contribution

It introduces a novel application of persistent homology to characterize structure-function relationships in flow networks, uncovering universal topological features linked to network functionality.

## Key findings

- Hidden topological sectors of uniform pressure identified
- Topological features strongly correlate with network responses
- Universal topological description applies across diverse networks

## Abstract

Complex networks encountered in biology are often characterized by significant structural diversity. Whether it be differences in the three-dimensional structure of allosteric proteins, or the variation among the micro-scale structures of organisms' cerebral vasculature systems, identifying relationships between structure and function often poses a difficult challenge. Here we showcase an approach to characterizing structure-function relationships in complex networks applied in the context of flow networks tuned to perform specific functions. Using persistent homology, we analyze flow networks tuned to perform complex multifunctional tasks, answering the question of how local changes in the network structure coordinate to create functionality at at the scale of the entire network. We find that the response of such networks encodes hidden topological features - sectors of uniform pressure - that are not apparent in the underlying network architectures, Regardless of differences in local connectivity, these features provide a universal topological description for all networks that perform these types of functions. We show that these features correlate strongly with the tuned response, providing a clear topological relationship between structure and function and structural insight into the limits of multifunctionality.