The hidden topological structure of flow network functionality

TL;DR

This paper reveals a hidden topological structure in flow networks, showing how sectors of uniform pressure encode network functionality and relate to the network's ability to perform complex tasks.

Contribution

It introduces a novel topological analysis using persistent homology to connect microscopic structural changes with macroscopic network functions.

Findings

Networks encode sectors of uniform pressure not visible in architecture.

These sectors correlate strongly with network functions.

Topological structure bounds the complexity of achievable tasks.

Abstract

The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms can robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function of these networks: networks with different local architectures can perform the same functions. Here we answer the question of how structural changes at the microscopic level are able to collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex multifunctional tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of uniform pressure. Although these sectors are not apparent in the underlying network architectures, we find that they nonetheless correlate strongly with the tuned function. We conclude that the connectivity of these sectors, rather than that of the individual nodes, provides a quantitative relationship between structure and function in flow networks. Finally, we use this topological description to place a bound on the limits of task complexity.