Hierarchical Ordering of Reticular Networks
Publication Date
June 06, 2012
Journal
PLOS ONE
Authors
Yuriy Mileyko, Herbert Edelsbrunner, Charles A. Price & Joshua S. Weitz
Volume
7
Issue
6
Pages
e36715
DOI
https://dx.plos.org/10.1371/journal.pone.0036715
Publisher URL
http://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0036715
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/22701559
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3368924
Europe PMC
http://europepmc.org/abstract/MED/22701559
Web of Science
000305348400006
Scopus
84862006719
Mendeley
http://www.mendeley.com/research/hierarchical-ordering-reticular-networks
Events
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Mendeley | Further Information

{"title"=>"Hierarchical ordering of reticular networks", "type"=>"journal", "authors"=>[{"first_name"=>"Yuriy", "last_name"=>"Mileyko", "scopus_author_id"=>"16310337000"}, {"first_name"=>"Herbert", "last_name"=>"Edelsbrunner", "scopus_author_id"=>"7005273880"}, {"first_name"=>"Charles A.", "last_name"=>"Price", "scopus_author_id"=>"8651573400"}, {"first_name"=>"Joshua S.", "last_name"=>"Weitz", "scopus_author_id"=>"7102347776"}], "year"=>2012, "source"=>"PLoS ONE", "identifiers"=>{"issn"=>"19326203", "arxiv"=>"arXiv:1110.1413v1", "scopus"=>"2-s2.0-84862006719", "pui"=>"364979279", "doi"=>"10.1371/journal.pone.0036715", "isbn"=>"1932-6203", "sgr"=>"84862006719", "pmid"=>"22701559"}, "id"=>"d0d6bdda-9578-323b-a0b4-2e735324c2a5", "abstract"=>"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.", "link"=>"http://www.mendeley.com/research/hierarchical-ordering-reticular-networks", "reader_count"=>29, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>1, "Researcher"=>5, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>1, "Student > Master"=>2, "Student > Bachelor"=>1, "Professor"=>2}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>1, "Researcher"=>5, "Student > Doctoral Student"=>3, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>1, "Student > Master"=>2, "Student > Bachelor"=>1, "Professor"=>2}, "reader_count_by_subject_area"=>{"Engineering"=>1, "Unspecified"=>1, "Environmental Science"=>1, "Agricultural and Biological Sciences"=>10, "Medicine and Dentistry"=>1, "Physics and Astronomy"=>6, "Computer Science"=>3, "Earth and Planetary Sciences"=>6}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>1}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>6}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>6}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>10}, "Computer Science"=>{"Computer Science"=>3}, "Unspecified"=>{"Unspecified"=>1}, "Environmental Science"=>{"Environmental Science"=>1}}, "reader_count_by_country"=>{"United States"=>1, "Finland"=>1, "Brazil"=>1, "Germany"=>1}, "group_count"=>0}

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/627190"], "description"=>"<p>The levels are assigned to the loops and branches of the network from <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036715#pone-0036715-g001\" target=\"_blank\">Fig. 1C</a>. Edge levels are shown on the left, where black edges have order 1, light blue edges have order 2 and gold edges have order 3; reticular edges are dashed. Face levels are shown in the co-tree on the right, where white nodes have order 1, light blue nodes have order 2 and gold nodes have order 3. Leaves of the co-tree are labeled by the corresponding faces while other nodes are labeled by the reticular edges causing the merger of the two child nodes. Numbers are Tokunaga statistics for the spanning tree and indicate the number of edges of level <i>j</i> joining with edges of level <i>i </i><a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036715#pone.0036715-Tokunaga1\" target=\"_blank\">[43]</a>. Similarly, are Tokunaga statistics for the reticulate co-tree and indicate the number of edges and faces of level <i>j</i> merging with edges of level <i>i</i>. For both <i>M</i> and <i>N</i>, statistics are only collected when .</p>", "links"=>[], "tags"=>["hierarchical"], "article_id"=>297683, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g003", "stats"=>{"downloads"=>1, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_hierarchical_levels_/297683", "title"=>"Example of hierarchical levels.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:11:18"}
  • {"files"=>["https://ndownloader.figshare.com/files/627493"], "description"=>"<p>(A) The part of the network containing the two edges being transposed. The brown, blue, and green triangles (and edges) denote the subtrees adjacent to the edges. (B) The effect of the transposition on the structure of the spanning tree and its hierarchical levels.</p>", "links"=>[], "tags"=>["transposition", "reticular"], "article_id"=>297988, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g006", "stats"=>{"downloads"=>1, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_a_single_transposition_of_a_reticular_edge_and_a_tree_edge_/297988", "title"=>"Effect of a single transposition of a reticular edge and a tree edge.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:12:58"}
  • {"files"=>["https://ndownloader.figshare.com/files/627423"], "description"=>"<p>(A) the part of the network containing the two edges being transposed and the effect of the transposition on the structure of the co-tree; (B) possible level changes caused by the transposition.</p>", "links"=>[], "tags"=>["transposition", "reticular"], "article_id"=>297911, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g005", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effect_of_a_single_transposition_of_two_reticular_edges_/297911", "title"=>"Effect of a single transposition of two reticular edges.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:12:36"}
  • {"files"=>["https://ndownloader.figshare.com/files/627337"], "description"=>"<p>The network has <i>m</i> faces, where for some integer . of these faces are adjacent squares and the other one is the unbounded face. Vertical edges are removed before horizontal edges as follows: (A) The edges are removed sequentially from left to right. The corresponding co-tree has the shape of a “comb” and the maximal hierarchical level is 2; (B) The edges are removed from left to right skipping every second edge. The process is repeated until all vertical edges except the rightmost one are removed. The corresponding co-tree has the height which is the maximal hierarchical level.</p>", "links"=>[], "tags"=>["cases"], "article_id"=>297832, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g004", "stats"=>{"downloads"=>1, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_the_two_extreme_cases_of_the_loop_hierarchy_/297832", "title"=>"Example of the two extreme cases of the loop hierarchy.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:12:07"}
  • {"files"=>["https://ndownloader.figshare.com/files/626930"], "description"=>"<p>A common “root” or outlet denoted by the red dot at the bottom of each network: (A) Horton-Strahler stream order of branch hierarchy in a tree network; (B) Reticular network with possible loop hierarchy: the, blue, dashed loop might be less important than the red, dotted loop; (C) Reticular network of (B) with weights.</p>", "links"=>[], "tags"=>["networks", "hierarchical"], "article_id"=>297418, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g001", "stats"=>{"downloads"=>1, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Examples_of_networks_with_hierarchical_structure_/297418", "title"=>"Examples of networks with hierarchical structure.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:09:49"}
  • {"files"=>["https://ndownloader.figshare.com/files/627025"], "description"=>"<p>The merging is applied to the graph from <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036715#pone-0036715-g001\" target=\"_blank\">Fig. 1C</a>. Red, dashed edges are the ones removed during merging, the corresponding numbers show their levels. Levels of faces is encoded by the color: white faces have level 1, light blue faces have level 2, and gold faces have level 3. Note that is the unbounded face.</p>", "links"=>[], "tags"=>["merging"], "article_id"=>297519, "categories"=>["Information And Computing Sciences", "Mathematics", "Biological Sciences", "Physics", "Plant Biology"], "users"=>["Yuriy Mileyko", "Herbert Edelsbrunner", "Charles A. Price", "Joshua S. Weitz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0036715.g002", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_An_illustration_of_the_loop_merging_procedure_/297519", "title"=>"An illustration of the loop merging procedure.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 04:10:26"}

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Relative Metric

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