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Quantifying loopy network architectures.

Katifori E, Magnasco MO - PLoS ONE (2012)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Center for Studies in Physics and Biology, The Rockefeller University, New York, New York, United States of America. ekatifori@rockefeller.edu

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 approaches 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.

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Variability in natural loopy networks.(a), (b) Leaf vasculature of two dicotyledonous species. (c) Detail of leaf collected from the same plant as leaf (a). The venation of (a) and (c) is predominately reticulate, (b) is percurrent. In general, leaves from the same plant (or species) share statistically similar architectural properties, as compared to leaves from different species. The scale is 1 cm.
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pone-0037994-g001: Variability in natural loopy networks.(a), (b) Leaf vasculature of two dicotyledonous species. (c) Detail of leaf collected from the same plant as leaf (a). The venation of (a) and (c) is predominately reticulate, (b) is percurrent. In general, leaves from the same plant (or species) share statistically similar architectural properties, as compared to leaves from different species. The scale is 1 cm.

Mentions: Many biological distribution and structural networks contain dense numbers of reentrant loops. The venation of angiosperm leaves (Fig. 1) [1], the structural veins of insect wings, the continuously adapting foraging networks of some fungi and slime molds [2], the vasculature of animal organs such as the adrenal glands, the brain [3] and the liver are just a few of a large number of examples where physical networks developed loops in living organisms. These networks perform functions crucial to the survival of the organisms that use them. The hierarchical organization and the intricacies of the architecture of these highly interconnected networks dictate the efficacy in providing support or distributing load under varying conditions. In some cases the function of closed loops and how many there should be is intuitively obvious; the webbing-like veins of a dragonfly wing have cross-bracings that serve to maintain rigidity and resistance with a minimum of weight. In other cases it is not self-evident why there are as many loops as observed.


Quantifying loopy network architectures.

Katifori E, Magnasco MO - PLoS ONE (2012)

Variability in natural loopy networks.(a), (b) Leaf vasculature of two dicotyledonous species. (c) Detail of leaf collected from the same plant as leaf (a). The venation of (a) and (c) is predominately reticulate, (b) is percurrent. In general, leaves from the same plant (or species) share statistically similar architectural properties, as compared to leaves from different species. The scale is 1 cm.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3368948&req=5

pone-0037994-g001: Variability in natural loopy networks.(a), (b) Leaf vasculature of two dicotyledonous species. (c) Detail of leaf collected from the same plant as leaf (a). The venation of (a) and (c) is predominately reticulate, (b) is percurrent. In general, leaves from the same plant (or species) share statistically similar architectural properties, as compared to leaves from different species. The scale is 1 cm.
Mentions: Many biological distribution and structural networks contain dense numbers of reentrant loops. The venation of angiosperm leaves (Fig. 1) [1], the structural veins of insect wings, the continuously adapting foraging networks of some fungi and slime molds [2], the vasculature of animal organs such as the adrenal glands, the brain [3] and the liver are just a few of a large number of examples where physical networks developed loops in living organisms. These networks perform functions crucial to the survival of the organisms that use them. The hierarchical organization and the intricacies of the architecture of these highly interconnected networks dictate the efficacy in providing support or distributing load under varying conditions. In some cases the function of closed loops and how many there should be is intuitively obvious; the webbing-like veins of a dragonfly wing have cross-bracings that serve to maintain rigidity and resistance with a minimum of weight. In other cases it is not self-evident why there are as many loops as observed.

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Center for Studies in Physics and Biology, The Rockefeller University, New York, New York, United States of America. ekatifori@rockefeller.edu

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 approaches 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.

Show MeSH