Hierarchical ordering of reticular networks.
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In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations.In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges.We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.
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Affiliation: Department of Mathematics, Duke University, Durham, North Carolina, United States of America. yury@math.duke.edu
ABSTRACT
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The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks. Related in: MedlinePlus |
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Mentions: The construction of removes edges from G which are responsible for the existence of loops. We shall call such edges reticular. Assignment of levels for such edges is based on the assumption that a merger should not be more significant than any of the merging elements. Notice that after removing reticular edges from G we have a spanning tree of G, which we denote by . This tree captures the tree-like structure of the original network, and we can assign hierarchical levels to its edges using the original Horton-Strahler algorithm. We only need to determine which vertex should be the root, and we do this by finding the vertex with a single incident weight of maximum weight. Hence, as noted in the Methods, the final step is to aply the Horton-Strahler ordering to the remainder of the graph (which is a rooted tree). The result of the complete algorithm applied to the tree in Fig. 1C is provided in Fig. 3. |
View Article: PubMed Central - PubMed
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina, United States of America. yury@math.duke.edu