Topological Data Analysis of Biological Aggregation Models
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{"title"=>"Topological data analysis of biological aggregation models", "type"=>"journal", "authors"=>[{"first_name"=>"Chad M.", "last_name"=>"Topaz", "scopus_author_id"=>"6508146488"}, {"first_name"=>"Lori", "last_name"=>"Ziegelmeier", "scopus_author_id"=>"55834559700"}, {"first_name"=>"Tom", "last_name"=>"Halverson", "scopus_author_id"=>"6701677255"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"pui"=>"604377725", "sgr"=>"84929353182", "issn"=>"19326203", "arxiv"=>"1412.6430", "pmid"=>"25970184", "scopus"=>"2-s2.0-84929353182", "doi"=>"10.1371/journal.pone.0126383"}, "id"=>"12e79500-45a7-396e-a274-d278a4468023", "abstract"=>"We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.", "link"=>"http://www.mendeley.com/research/topological-data-analysis-biological-aggregation-models", "reader_count"=>54, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>3, "Researcher"=>10, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>3, "Student > Master"=>7, "Other"=>5, "Student > Bachelor"=>4, "Lecturer"=>2, "Professor"=>4}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>3, "Researcher"=>10, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>13, "Student > Postgraduate"=>3, "Student > Master"=>7, "Other"=>5, "Student > Bachelor"=>4, "Lecturer"=>2, "Professor"=>4}, "reader_count_by_subject_area"=>{"Unspecified"=>3, "Engineering"=>6, "Biochemistry, Genetics and Molecular Biology"=>1, "Mathematics"=>14, "Agricultural and Biological Sciences"=>4, "Philosophy"=>1, "Neuroscience"=>2, "Physics and Astronomy"=>9, "Psychology"=>1, "Social Sciences"=>1, "Computer Science"=>11, "Economics, Econometrics and Finance"=>1}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>6}, "Neuroscience"=>{"Neuroscience"=>2}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>9}, "Psychology"=>{"Psychology"=>1}, "Economics, Econometrics and Finance"=>{"Economics, Econometrics and Finance"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>4}, "Computer Science"=>{"Computer Science"=>11}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}, "Mathematics"=>{"Mathematics"=>14}, "Unspecified"=>{"Unspecified"=>3}, "Philosophy"=>{"Philosophy"=>1}}, "reader_count_by_country"=>{"Republic of Singapore"=>1, "Belgium"=>1, "United States"=>2, "Luxembourg"=>1, "Bulgaria"=>1}, "group_count"=>3}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2066285"], "description"=>"<p>These simulations are analogous to Fig. 1 in [<a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.ref011\" target=\"_blank\">11</a>]. Circles indicate particle positions and line segments represent heading. For all simulations, <i>N</i> = 300 particles, the particle speed is <i>v</i><sub>0</sub> = 0.03, and the initial state consists of uniform random positions and headings. We vary box size ℓ and noise <i>η</i>. Dotted lines indicate the bounds of the periodic domain. (A) Groups moving in different directions with ℓ = 25, <i>η</i> = 0.1, <i>t</i> = 3000. (B) Random movement with some correlation with ℓ = 7, <i>η</i> = 2, <i>t</i> = 600. (C) Highly polarized motion with ℓ = 5, <i>η</i> = 0.1, <i>t</i> = 300.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413549, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g007", "stats"=>{"downloads"=>2, "page_views"=>19, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Simulation_snapshots_of_the_Vicsek_model_9_/1413549", "title"=>"Simulation snapshots of the Vicsek model (9).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066284"], "description"=>"<p>The simulation that generated these data was seeded with the initial condition in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g005\" target=\"_blank\">Fig 5(A)</a> and a typical snapshot is shown in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g007\" target=\"_blank\">Fig 7(A)</a>. (A) Normalized average velocity order parameter <i>φ</i>(<i>t</i>). (B) Contour plot of Betti number </p><p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. (C) Contour plot of Betti number <p></p><p></p><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. The topological analysis reveals dynamics not captured by the order parameter, namely cluster formation and the loss of topological circles consistent with particles aligning and covering only one dimension of the periodic simulation domain. See text for a more comprehensive analysis.<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413548, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g006", "stats"=>{"downloads"=>0, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Aggregate_behavior_of_the_Vicsek_model_simulation_1_/1413548", "title"=>"Aggregate behavior of the Vicsek model, simulation #1.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066281"], "description"=>"<p>The blue 1-cycle and the red 1-cycle are homologous (equivalent), because their difference is the boundary of a triangle, shown in green; see text for a detailed explanation.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413545, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g003", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_homologous_cycles_/1413545", "title"=>"Example of homologous cycles.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066279"], "description"=>"<p>These <i>k</i>-simplices are the building blocks used to construct a simplicial complex from a point cloud of data.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413543, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g001", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Oriented_k_simplices_for_k_0_1_2_3_/1413543", "title"=>"Oriented <i>k</i>-simplices for <i>k</i> = 0, 1, 2, 3.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066280"], "description"=>"<p>The 18 points are 0-simplices. Two 0-simplices form a 1-simplex (an edge) if their <i>ɛ</i>/2-neighborhoods (yellow circles) intersect. Three vertices form a 2-simplex (a triangle) if they are pairwise connected by edges. Four vertices form a 3-simplex (a tetrahedron) if they are pairwise connected by edges.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413544, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g002", "stats"=>{"downloads"=>0, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_a_Vietoris_Rips_complex_/1413544", "title"=>"Example of a Vietoris-Rips complex.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066289"], "description"=>"<p>Circles indicate positions of the <i>N</i> = 500 particles in an unbounded plane, line segments represent heading, and blue (red) agents are traveling (counter)clockwise. Over time, the group develops a hollow core and a double-mill structure in which a majority of agents travel clockwise, but a minority persists in the counterclockwise orientation. (A) Time <i>t</i> = 5. (B) Time <i>t</i> = 23. (C) Time <i>t</i> = 34. The other model parameters used in this simulation are <i>α</i> = 1.5, <i>β</i> = 0.5, <i>C</i><sub><i>r</i></sub> = 1, <i>L</i><sub><i>r</i></sub> = 0.5, <i>C</i><sub><i>a</i></sub> = 0.5, <i>L</i><sub><i>a</i></sub> = 2.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413553, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g011", "stats"=>{"downloads"=>2, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Simulation_snapshots_of_the_D_8217_Orsogna_model_11_/1413553", "title"=>"Simulation snapshots of the D’Orsogna model (11).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066287"], "description"=>"<p>A typical snapshot is shown in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g007\" target=\"_blank\">Fig 7(B)</a>. (A) Normalized average velocity order parameter <i>φ</i>(<i>t</i>). (B) Contour plot of Betti number </p><p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. (C) Contour plot of Betti number <p></p><p></p><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. The topological analysis suggests sporadic coagulation and fragmentation of short-lived clusters, and the loss of a topological circle, consistent with particles aligning. See text for a more comprehensive analysis.<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413551, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g009", "stats"=>{"downloads"=>0, "page_views"=>19, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Aggregate_behavior_of_the_Vicsek_model_simulation_2_/1413551", "title"=>"Aggregate behavior of the Vicsek model, simulation #2.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066288"], "description"=>"<p>A typical snapshot is shown in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g007\" target=\"_blank\">Fig 7(C)</a>. (A) Normalized average velocity order parameter <i>φ</i>(<i>t</i>). (B) Contour plot of Betti number </p><p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. (C) Contour plot of Betti number <p></p><p></p><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. The topological analysis shows essentially no cluster formation; the narrow region in which <p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn><p></p><p></p> arises from an isolated agent. The two persistent topological circles are consistent with highly aligned particles covering both dimensions of the periodic spatial domain. Topological features become fairly stagnant once the entire group forms a large, aligned cluster traveling as a rigid body early in the simulation. See text for a more comprehensive analysis.<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413552, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g010", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Aggregate_behavior_of_the_Vicsek_model_simulation_3_/1413552", "title"=>"Aggregate behavior of the Vicsek model, simulation #3.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066286"], "description"=>"<p>These states correspond to the dashed vertical bars in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g006\" target=\"_blank\">Fig 6(B)</a>. (A) Time <i>t</i> = 2080. (B) Time <i>t</i> = 2150. The topological signature in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g006\" target=\"_blank\">Fig 6(B)</a> picks up subtle differences between these states. For panel (A) here, there are ranges of the persistence parameter <i>ɛ</i> over which one observes four, three, and two connected components before coalescing into one. For (B), the transition from four connected components to two happens over a much smaller range of <i>ɛ</i> because the two larger clusters on the right merge into one on approximately the same spatial scale that the two smaller clusters on the left do. The topological differences between (A) and (B) are not readily visible to the eye in the snapshots, nor are they reflected in the order parameter <i>φ</i>(<i>t</i>) in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g006\" target=\"_blank\">Fig 6(A)</a>.</p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413550, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g008", "stats"=>{"downloads"=>1, "page_views"=>21, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Snapshots_of_the_Vicsek_model_simulation_1_/1413550", "title"=>"Snapshots of the Vicsek model, simulation #1.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066283"], "description"=>"<p>(A) Random initial positions (<i>x</i>, <i>y</i>) and headings <i>θ</i> of <i>N</i> = 300 particles in a square of size ℓ = 25 with periodic boundary conditions. The underlying space in which the data lives is a three-torus </p><p></p><p></p><p>T<mn>3</mn></p><p></p><p></p> which has Betti numbers <p></p><p>b<mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo stretchy=\"false\">)</mo></p><p></p>. (B) Barcode for Betti number <p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, showing topological connected components. The zoomed box shows a single persistent bar, corresponding to the entire ensemble of particles. (C) Barcode for Betti number <p></p><p></p><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><p></p><p></p>, showing topological circles. The zoomed box shows three persistent bars, representing the three circles comprising the three-torus. (D) Persistence plot, which displays the information in (B) and (C) by encoding each bar’s starting and ending value of <i>ɛ</i> as a point in the Cartesian plane. Red points show <p></p><p></p><p>b<mn>0</mn></p><p></p><p></p> and blue points show <p></p><p></p><p>b<mn>1</mn></p><p></p><p></p>. The zoomed box shows the three points representing the three persistent topological circles of the random initial condition in (A).<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413547, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g005", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_A_random_initial_condition_used_to_simulate_the_Vicsek_model_9_and_topological_analysis_of_this_initial_state_/1413547", "title"=>"A random initial condition used to simulate the Vicsek model (9) and topological analysis of this initial state.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066282"], "description"=>"<p>The top four figures display the simplicial complex of 18 points for different values of the proximity parameter <i>ɛ</i>. The vertical lines in the barcode correspond to these four levels of <i>ɛ</i>. The number of horizontal bars intersecting each line give the values of </p><p></p><p>b<mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo stretchy=\"false\">)</mo><mo>,</mo><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><p></p><p></p>. For the parameters selected, <p></p><p>b<mo stretchy=\"false\">(</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mn>18</mn><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo></p><p></p>, <p></p><p>b<mo stretchy=\"false\">(</mo><mn>5</mn><mo>.</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mn>11</mn><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>,</mo></p><p></p><p></p><p>b<mo stretchy=\"false\">(</mo><mn>7</mn><mo>.</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo>,</mo></p><p></p> and <p></p><p>b<mo stretchy=\"false\">(</mo><mn>9</mn><mo>.</mo><mn>5</mn><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></p><p></p>. See text for further discussion.<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413546, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g004", "stats"=>{"downloads"=>6, "page_views"=>94, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_of_the_topological_barcode_of_a_Vietoris_Rips_complex_/1413546", "title"=>"Example of the topological barcode of a Vietoris-Rips complex.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/2066290"], "description"=>"<p>Snapshot of the time evolution are shown in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126383#pone.0126383.g011\" target=\"_blank\">Fig 11</a>. (A) Three order parameters: polarization <i>P</i> (red), angular momentum <i>M</i> (green), and absolute angular momentum <i>M</i><sub><i>abs</i></sub> (blue). (B) Contour plot of Betti number </p><p></p><p></p><p>b<mn>0</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. (C) Contour plot of Betti number <p></p><p></p><p>b<mn>1</mn></p><mo stretchy=\"false\">(</mo><mi>ɛ</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><p></p><p></p>. At times below <i>t</i> ≈ 20, there is little topological structure. For <i>t</i> > 20, we have one or—intermittently—two connected components of data points. There are two discernible topological circles for smaller <i>ɛ</i> and one circle for larger <i>ɛ</i>. These circles survive for long periods of simulation time. The topological signature of the first two Betti numbers, <p></p><p>b<mo>=</mo><mo stretchy=\"false\">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></p><p></p>, is consistent with a double mill structure. See text for a more comprehensive analysis.<p></p>", "links"=>[], "tags"=>["topological persistence scale", "Biological Aggregation Models", "displays Betti numbers", "order parameters", "simulation time frame", "Betti numbers count", "topological data analysis", "model"], "article_id"=>1413554, "categories"=>["Uncategorised"], "users"=>["Chad M. Topaz", "Lori Ziegelmeier", "Tom Halverson"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0126383.g012", "stats"=>{"downloads"=>2, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Aggregate_behavior_of_the_D_8217_Orsogna_model_/1413554", "title"=>"Aggregate behavior of the D’Orsogna model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-13 03:46:31"}

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  • {"unique-ip"=>"11", "full-text"=>"9", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"19", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"1"}
  • {"unique-ip"=>"8", "full-text"=>"7", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"3"}
  • {"unique-ip"=>"7", "full-text"=>"9", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"1"}
  • {"unique-ip"=>"10", "full-text"=>"6", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"4"}
  • {"unique-ip"=>"4", "full-text"=>"5", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"9"}
  • {"unique-ip"=>"9", "full-text"=>"8", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"5"}
  • {"unique-ip"=>"9", "full-text"=>"6", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"6"}
  • {"unique-ip"=>"9", "full-text"=>"8", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"7"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"10"}
  • {"unique-ip"=>"9", "full-text"=>"7", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"8"}
  • {"unique-ip"=>"13", "full-text"=>"14", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"11"}
  • {"unique-ip"=>"8", "full-text"=>"8", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"12"}
  • {"unique-ip"=>"10", "full-text"=>"11", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"2"}
  • {"unique-ip"=>"5", "full-text"=>"6", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"3"}
  • {"unique-ip"=>"7", "full-text"=>"7", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"4"}
  • {"unique-ip"=>"8", "full-text"=>"9", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"5"}
  • {"unique-ip"=>"11", "full-text"=>"7", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"8"}
  • {"unique-ip"=>"14", "full-text"=>"17", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"9"}
  • {"unique-ip"=>"12", "full-text"=>"10", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"10"}
  • {"unique-ip"=>"11", "full-text"=>"12", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"12"}

Relative Metric

{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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