Mileyko, Yuriy ; Edelsbrunner, HerbertIST Austria ; Price, Charles ; Weitz, Joshua
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.
his work was supported by the National Science Foundation Plant Genome Research Program (grant 0820624 to H.E. and J.S.W.), the Defense Advanced Projects Research Agency (grant HR0011-09-1-0055 to H.E. and J.S.W.), and the European Science Foundation (under the Research Networking Programme on “Applied and Computational Algebraic Topology” run by H.E.). Joshua S. Weitz, Ph.D., holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund. During preparation of this manuscript the authors became aware of a related work by Katifori and Magnasco (arXiv:1110.1412v1), concurrently submitted and accepted for publication in PLoS ONE.
Mileyko Y, Edelsbrunner H, Price C, Weitz J. Hierarchical ordering of reticular networks. PLoS One. 2012;7(6). doi:10.1371/journal.pone.0036715
Mileyko, Y., Edelsbrunner, H., Price, C., & Weitz, J. (2012). Hierarchical ordering of reticular networks. PLoS One, 7(6). https://doi.org/10.1371/journal.pone.0036715
Mileyko, Yuriy, Herbert Edelsbrunner, Charles Price, and Joshua Weitz. “Hierarchical Ordering of Reticular Networks.” PLoS One 7, no. 6 (2012). https://doi.org/10.1371/journal.pone.0036715.
Y. Mileyko, H. Edelsbrunner, C. Price, and J. Weitz, “Hierarchical ordering of reticular networks,” PLoS One, vol. 7, no. 6, 2012.
Mileyko Y, Edelsbrunner H, Price C, Weitz J. 2012. Hierarchical ordering of reticular networks. PLoS One. 7(6).
Mileyko, Yuriy, et al. “Hierarchical Ordering of Reticular Networks.” PLoS One, vol. 7, no. 6, e36715, Public Library of Science, 2012, doi:10.1371/journal.pone.0036715.
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