---
_id: '3159'
abstract:
- lang: eng
  text: 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.
acknowledgement: "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.\r\n\r\n\r\n\r\nDuring
  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."
article_number: e36715
article_processing_charge: No
author:
- first_name: Yuriy
  full_name: Mileyko, Yuriy
  last_name: Mileyko
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Charles
  full_name: Price, Charles
  last_name: Price
- first_name: Joshua
  full_name: Weitz, Joshua
  last_name: Weitz
citation:
  ama: Mileyko Y, Edelsbrunner H, Price C, Weitz J. Hierarchical ordering of reticular
    networks. <i>PLoS One</i>. 2012;7(6). doi:<a href="https://doi.org/10.1371/journal.pone.0036715">10.1371/journal.pone.0036715</a>
  apa: Mileyko, Y., Edelsbrunner, H., Price, C., &#38; Weitz, J. (2012). Hierarchical
    ordering of reticular networks. <i>PLoS One</i>. Public Library of Science. <a
    href="https://doi.org/10.1371/journal.pone.0036715">https://doi.org/10.1371/journal.pone.0036715</a>
  chicago: Mileyko, Yuriy, Herbert Edelsbrunner, Charles Price, and Joshua Weitz.
    “Hierarchical Ordering of Reticular Networks.” <i>PLoS One</i>. Public Library
    of Science, 2012. <a href="https://doi.org/10.1371/journal.pone.0036715">https://doi.org/10.1371/journal.pone.0036715</a>.
  ieee: Y. Mileyko, H. Edelsbrunner, C. Price, and J. Weitz, “Hierarchical ordering
    of reticular networks,” <i>PLoS One</i>, vol. 7, no. 6. Public Library of Science,
    2012.
  ista: Mileyko Y, Edelsbrunner H, Price C, Weitz J. 2012. Hierarchical ordering of
    reticular networks. PLoS One. 7(6), e36715.
  mla: Mileyko, Yuriy, et al. “Hierarchical Ordering of Reticular Networks.” <i>PLoS
    One</i>, vol. 7, no. 6, e36715, Public Library of Science, 2012, doi:<a href="https://doi.org/10.1371/journal.pone.0036715">10.1371/journal.pone.0036715</a>.
  short: Y. Mileyko, H. Edelsbrunner, C. Price, J. Weitz, PLoS One 7 (2012).
date_created: 2018-12-11T12:01:44Z
date_published: 2012-06-06T00:00:00Z
date_updated: 2025-09-30T07:53:26Z
day: '06'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1371/journal.pone.0036715
external_id:
  isi:
  - '000305348400006'
file:
- access_level: open_access
  checksum: 515a98ad72e470752f03f13663dcaff8
  content_type: application/pdf
  creator: kschuh
  date_created: 2019-02-05T12:38:43Z
  date_updated: 2020-07-14T12:46:01Z
  file_id: '5922'
  file_name: 2012_PLoS_Mileyko.PDF
  file_size: 541583
  relation: main_file
file_date_updated: 2020-07-14T12:46:01Z
has_accepted_license: '1'
intvolume: '         7'
isi: 1
issue: '6'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '06'
oa: 1
oa_version: Published Version
publication: PLoS One
publication_status: published
publisher: Public Library of Science
publist_id: '3530'
pubrep_id: '385'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Hierarchical ordering of reticular networks
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 7
year: '2012'
...