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

Abstract

Complex networks encountered in biology are often characterized by significant structural diversity. Whether due to differences in the three-dimensional structure of allosteric proteins, or the variation among the microscale 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 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.

Demonstration of Decentralized, Physics-Driven Learning

“…Allostery is a common feature of proteins [27], in which an input signal, namely strain applied to a local region of the protein by binding a regulatory molecule, gives rise to a desired strain or conformational change elsewhere in the protein, enabling or preventing binding of a substrate molecule. In a related problem of 'flow allostery' [29,42,43], a pressure drop in one region of a flow network, (e.g. across input arteries in the brain vascular network) gives rise to desired pressure drops elsewhere in the brain at designated output locations that can be quite distant from the input arteries, allowing the vascular system to deliver enhanced blood flow and therefore more oxygen to active parts of the brain.…”

“…This is why the system can learn new tasks. There is still much to be understood about even our modest 16-edge system, but the simplicity of its local rules and basis in well-understood physical laws suggest the possibility of understanding exactly what and how it learns [42,43,46]. Certainly theoretical understanding seems less difficult to attain for the physical learning machine than for many neuromorphic realizations, not to mention the brain itself.…”

Structural analysis of bio-inspired spatial systems: network efficiency of fractal morphologies

“…One characteristic of these systems is their physical fractality, often quantified by the fractal dimension [1,11,12,13]. However, although the fractal dimension is a good measure of morphological complexity, it does not provide a comprehensive account of the micro structural features that could make branched morphologies relevant at the macro level for the system's biological function [4,14,15,16]. Therefore, here we considered the structural analysis of bio-inspired spatial systems based on fractal and network theory approaches in order to identify the morphological and topological features that could make tree-like systems better at exploring space under limited amounts of matter, energy and information.…”

Extremal lifetimes of persistent cycles

“…The simple and effective idea is to leverage invariants from algebraic topology to extract insights from data. While initially TDA started off as an eccentric mathematical idea, it is now applied in a wide variety of disciplines, ranging from astronomy and biology to finance and materials science Gidea and Katz (2018); Pranav et al (2016); Rocks et al (2020); Saadatfar et al (2017).…”