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The loopy leaf4/19/2023 ![]() To compare naturally occurring networks with the computer simulated models we therefore need to be able to test the null hypothesis that the two networks in question have been drawn from the same distribution. Similarly, networks produced in silico by optimization routines or developmental simulations that incorporate the effects of biological noise exhibit architectures that are to some extent random: each simulation repeat will produce statistically similar, but never identical, networks –. For example, one can frequently determine by mere inspection of the leaf venation patterns if two leaves are specimens from two different species ( Fig. These networks are the result of developmental processes that frequently dictate not the exact position of each network edge but the overall organization in a statistical sense. Various reasons for the evolution of loopiness in biological distribution networks have been proposed –. In many cases, such as leaf venation, loopy networks evolved gradually from a tree architecture. In general, leaves from the same plant (or species) share statistically similar architectural properties, as compared to leaves from different species. The venation of (a) and (c) is predominately reticulate, (b) is percurrent. ![]() (c) Detail of leaf collected from the same plant as leaf (a). (a), (b) Leaf vasculature of two dicotyledonous species. In other cases it is not self-evident why there are as many loops as observed. 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. 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. These networks perform functions crucial to the survival of the organisms that use them. 1), the structural veins of insect wings, the continuously adapting foraging networks of some fungi and slime molds, the vasculature of animal organs such as the adrenal glands, the brain and the liver are just a few of a large number of examples where physical networks developed loops in living organisms. Many biological distribution and structural networks contain dense numbers of reentrant loops. Besides their evident technological importance, these networks are central to the function of living beings because of their concrete physicality they are sometimes far more accessible to experimental analysis than other important biological networks, and hence offer an important window into the organization and function of naturally evolved large-scale networks. To some extent, structural load-bearing networks can also be considered in this category, as their job is the distribution of stress-strain. 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.Īmong the many different classes of complex systems that can primarily be described as “networks”, an important subclass concerns physical networks devoted to transportation of various entities, such as fluids or energy. 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. 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 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. ![]() 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. 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. Biology presents many examples of planar distribution and structural networks having dense sets of closed loops. ![]()
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