Quantifying loopy network architectures

TL;DR

This paper introduces a hierarchical loop decomposition algorithm to quantify and compare the architectural organization of loopy networks, such as leaf vasculature and brain vasculature, by mapping them to binary trees and analyzing their structural metrics.

Contribution

The paper presents a novel algorithmic framework that maps loopy networks to binary trees, enabling quantitative analysis of their hierarchical organization and comparison with theoretical models.

Findings

The framework successfully characterizes the hierarchical structure of natural and artificial networks.

Metrics derived from the trees reveal differences in architectural organization.

The method decouples geometric and topological information for detailed analysis.

Abstract

Biology presents many examples of planar distribution and structural networks having dense sets of closed loops. An archetype of this form of network organization is the vasculature of dicotyledonous leaves, which showcases a hierarchically-nested architecture containing closed loops at many different levels. Although a number of methods have been proposed to measure aspects of the structure of such networks, a robust metric to quantify their hierarchical organization is still lacking. We present an algorithmic framework, the hierarchical loop decomposition, that allows mapping loopy networks to binary trees, preserving in the connectivity of the trees the architecture of the original graph. We apply this framework to investigate computer generated graphs, such as artificial models and optimal distribution networks, as well as natural graphs extracted from digitized images of dicotyledonous leaves and vasculature of rat cerebral neocortex. We calculate various metrics based on the Asymmetry, the cumulative size distribution and the Strahler bifurcation ratios of the corresponding trees and discuss the relationship of these quantities to the architectural organization of the original graphs. This algorithmic framework decouples the geometric information (exact location of edges and nodes) from the metric topology (connectivity and edge weight) and it ultimately allows us to perform a quantitative statistical comparison between predictions of theoretical models and naturally occurring loopy graphs.