How Much Is the Whole Really More than the Sum of Its Parts? 1 ⊞ 1 = 2.5: Superlinear Productivity in Collective Group Actions
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{"title"=>"How much is the whole really more than the sum of its parts? 1+1 = 2.5: Superlinear productivity in collective group actions", "type"=>"journal", "authors"=>[{"first_name"=>"Didier", "last_name"=>"Sornette", "scopus_author_id"=>"7102654290"}, {"first_name"=>"Thomas", "last_name"=>"Maillart", "scopus_author_id"=>"25722210300"}, {"first_name"=>"Giacomo", "last_name"=>"Ghezzi", "scopus_author_id"=>"25929124600"}], "year"=>2014, "source"=>"PLoS ONE", "identifiers"=>{"arxiv"=>"1405.4298", "sgr"=>"84905368050", "pmid"=>"25084157", "scopus"=>"2-s2.0-84905368050", "issn"=>"19326203", "pui"=>"373686044", "doi"=>"10.1371/journal.pone.0103023"}, "id"=>"d1f19936-131f-3438-a52f-cc874714f187", "abstract"=>"In a variety of open source software projects, we document a superlinear growth of production ($R \\sim c^\\beta$) as a function of the number of active developers $c$, with $\\beta \\simeq 4/3$ with large dispersions. For a typical project in this class, doubling of the group size multiplies typically the output by a factor $2^\\beta=2.5$, explaining the title. This superlinear law is found to hold for group sizes ranging from 5 to a few hundred developers. We propose two classes of mechanisms, {\\it interaction-based} and {\\it large deviation}, along with a cascade model of productive activity, which unifies them. In this common framework, superlinear productivity requires that the involved social groups function at or close to criticality, in the sense of a subtle balance between order and disorder. We report the first empirical test of the renormalization of the exponent of the distribution of the sizes of first generation events into the renormalized exponent of the distribution of clusters resulting from the cascade of triggering over all generation in a critical branching process in the non-meanfield regime. Finally, we document a size effect in the strength and variability of the superlinear effect, with smaller groups exhibiting widely distributed superlinear exponents, some of them characterizing highly productive teams. In contrast, large groups tend to have a smaller superlinearity and less variability.", "link"=>"http://www.mendeley.com/research/much-whole-really-more-sum-parts-11-25-superlinear-productivity-collective-group-actions", "reader_count"=>25, "reader_count_by_academic_status"=>{"Professor > Associate Professor"=>1, "Researcher"=>8, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>11, "Student > Master"=>2, "Other"=>2}, "reader_count_by_user_role"=>{"Professor > Associate Professor"=>1, "Researcher"=>8, "Student > Doctoral Student"=>1, "Student > Ph. D. Student"=>11, "Student > Master"=>2, "Other"=>2}, "reader_count_by_subject_area"=>{"Unspecified"=>3, "Engineering"=>3, "Environmental Science"=>1, "Mathematics"=>2, "Physics and Astronomy"=>4, "Psychology"=>1, "Social Sciences"=>2, "Computer Science"=>7, "Earth and Planetary Sciences"=>1, "Economics, Econometrics and Finance"=>1}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>3}, "Social Sciences"=>{"Social Sciences"=>2}, "Physics and Astronomy"=>{"Physics and Astronomy"=>4}, "Psychology"=>{"Psychology"=>1}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>1}, "Economics, Econometrics and Finance"=>{"Economics, Econometrics and Finance"=>1}, "Computer Science"=>{"Computer Science"=>7}, "Mathematics"=>{"Mathematics"=>2}, "Unspecified"=>{"Unspecified"=>3}, "Environmental Science"=>{"Environmental Science"=>1}}, "reader_count_by_country"=>{"Netherlands"=>1, "United States"=>2, "Luxembourg"=>1, "Japan"=>2, "Australia"=>1, "Estonia"=>1}, "group_count"=>1}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1618651"], "description"=>"<p>For clarity, the time series are represented in logarithmic scale and they have been smoothed with a rolling window of days. Over the whole project history, various epochs of productive activity can be found. The background grey areas indicate three levels of the productivity exponent defined by <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103023#pone.0103023.e036\" target=\"_blank\">equation (1</a>) (light grey for , grey for and dark grey for ) for time windows of 250 days. Blank areas show time windows for which could not be fitted, mainly because the numbers of active contributors (resp. commits) were strongly varying over these periods. In other words, it is possible that super linear production was occurring in these periods but we could not determined it.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "apache", "contributors"], "article_id"=>1124781, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g002", "stats"=>{"downloads"=>4, "page_views"=>32, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Typical_time_series_of_open_source_software_development_e_g_Apache_Web_Server_with_active_contributors_green_area_and_their_productive_activity_red_area_/1124781", "title"=>"Typical time series of open source software development (e.g. Apache Web Server) with active contributors (green area) and their productive activity (red area).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618650"], "description"=>"<p>The distribution follows approximately a power law with exponent , with an apparent deviation in the tail possibly resulting from an over-sampling bias of large projects. The bend down for small projects is likely the result of an under-sampling bias.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "sizes", "quantified"], "article_id"=>1124780, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g001", "stats"=>{"downloads"=>1, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Distribution_of_project_sizes_in_our_sample_quantified_by_their_total_number_of_developers_/1124780", "title"=>"Distribution of project sizes in our sample quantified by their total number of developers.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618676"], "description"=>"<p>(<b>A</b>) superlinear exponent as a function of , the exponent of the power law tail distribution of first generation productivity for each of the days periods for which both values could be calibrated. The points are concentrated around with almost half of them ( over values) within the grey area delimited by and . To test for the relations and , we used a bi-Gaussian model. The dotted ellipses show the first three standard deviations around the barycenters and the black line represents the main axis with the bi-Gaussian model. We also performed a principal component analysis (PCA). The red dotted lines show the main direction of variance obtained with the PCA. Both methods show a positive relation between and only on second principal component (slope with PCA). (<b>B</b>) same as panel (A) for the dependence of versus with a concentration of points in the grey area (86 over 213 values) and . Both the bi-Gaussian fit and the PCA show strong evidence of a positive relation with slope with the bi-Gaussian approach and with the PCA.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology"], "article_id"=>1124807, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g007", "stats"=>{"downloads"=>3, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Verification_of_the_relationship_between_and_as_predicted_by_the_theory_/1124807", "title"=>"Verification of the relationship between , and as predicted by the theory.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618678"], "description"=>"<p>(<b>A</b>) Average superlinear exponent per project as a function of the cumulative number of contributors. The circle size reflects the number of exponents fitted per time window, for each project and entering the average statistics. The sampling ranges from (small disks) to (largest disk). exhibits a slightly negative slope as a function of ( and ). (<b>B</b>) To measure the prevalence of productive bursts in projects, we measure the ratio of periods with superlinear exponent over all periods for each project as a function of . We distinguish a cluster of points around and (i.e. contributors) with a positive relationship () of the ratio as a function of . Projects with a large pool of contributors () are more randomly scattered with a lower ratio and do not obey the same relationship, suggesting a different regime.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "superlinear", "exponent"], "article_id"=>1124808, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g008", "stats"=>{"downloads"=>4, "page_views"=>19, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Evolution_of_the_superlinear_exponent_as_a_function_of_project_size_/1124808", "title"=>"Evolution of the superlinear exponent as a function of project size.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618655"], "description"=>"<p>For the Apache Web Server project, the scaling exponent is (, ). For the vast majority of projects, the relation between lines of code and commits exhibits the same scaling with , suggesting that we can use either commits or lines of codes, as both provide a consistent and therefore robust measure of contribution (and in addition that commits may themselves result from cascades of code production.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "commits", "lines"], "article_id"=>1124785, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g003", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Scaling_relation_between_commits_and_lines_of_code_/1124785", "title"=>"Scaling relation between commits and lines of code.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618672"], "description"=>"<p>(<b>A</b>) (light blue) Triggering mechanism generating the clusters of size with renormalized exponent from the distribution of first generation “daughter events” with exponent . For the sake of simplicity, we represented one cluster of activity per contributor, but triggering can occur between contributors provided that the probability of triggering remains the same between all contributors. (<b>B</b>) (light green) shows how the triggering mechanism generates superlinear productive activity as a function of the number of active contributors .</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "superlinear", "cascading", "heavy-tailed", "distributions", "cumulative"], "article_id"=>1124802, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g006", "stats"=>{"downloads"=>1, "page_views"=>21, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Relationship_between_superlinear_productive_bursts_cascading_dynamics_and_heavy_tailed_distributions_of_1_st_generation_and_cumulative_contributions_/1124802", "title"=>"Relationship between superlinear productive bursts, cascading dynamics, and heavy-tailed distributions of 1<sup>st</sup> generation and cumulative contributions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618700", "https://ndownloader.figshare.com/files/1618701"], "description"=>"<div><p>In a variety of open source software projects, we document a superlinear growth of production intensity () as a function of the number of active developers , with a median value of the exponent , with large dispersions of from slightly less than up to . For a typical project in this class, doubling of the group size multiplies typically the output by a factor , explaining the title. This superlinear law is found to hold for group sizes ranging from 5 to a few hundred developers. We propose two classes of mechanisms, <i>interaction-based</i> and <i>large deviation</i>, along with a cascade model of productive activity, which unifies them. In this common framework, superlinear productivity requires that the involved social groups function at or close to criticality, or in a “superradiance” mode, in the sense of the appearance of a cooperative process and order involving a collective mode of developers defined by the build up of correlation between the contributions of developers. In addition, we report the first empirical test of the renormalization of the exponent of the distribution of the sizes of first generation events into the renormalized exponent of the distribution of clusters resulting from the cascade of triggering over all generation in a critical branching process in the non-meanfield regime. Finally, we document a size effect in the strength and variability of the superlinear effect, with smaller groups exhibiting widely distributed superlinear exponents, some of them characterizing highly productive teams. In contrast, large groups tend to have a smaller superlinearity and less variability.</p></div>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "superlinear"], "article_id"=>1124829, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>["https://dx.doi.org/10.1371/journal.pone.0103023.s001", "https://dx.doi.org/10.1371/journal.pone.0103023.s002"], "stats"=>{"downloads"=>3, "page_views"=>27, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_How_Much_Is_the_Whole_Really_More_than_the_Sum_of_Its_Parts_1_8862_1_8202_8202_2_5_Superlinear_Productivity_in_Collective_Group_Actions_/1124829", "title"=>"How Much Is the Whole Really More than the Sum of Its Parts? 1 ⊞ 1 = 2.5: Superlinear Productivity in Collective Group Actions", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618662"], "description"=>"<p>The scaling exponent ( and ) is shown as the slope of a straight line in double logarithmic scale. The error bars show the 25th and 75th percentiles of contributors log-bins.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "superlinear", "logarithmic", "contributors", "5-day", "windows", "apache", "server"], "article_id"=>1124792, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g004", "stats"=>{"downloads"=>3, "page_views"=>46, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Typical_superlinear_relation_in_double_logarithmic_scale_of_the_productive_contribution_as_a_function_of_active_contributors_per_5_day_time_windows_for_Apache_Web_Server_http_httpd_apache_org_/1124792", "title"=>"Typical superlinear relation in double logarithmic scale of the productive contribution as a function of active contributors per 5-day time windows for Apache Web Server (http://httpd.apache.org/).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618669"], "description"=>"<p>(<b>red circles</b>) Complementary cumulative distribution of all contributions per developer over a long period of time. is equivalent to measuring the cluster sizes of contributions following critical cascades (7). All distributions have been fitted using the maximum likelihood estimator (MLE). The distribution of cascade size is characterized by the exponent compared to the first generation daughter events distribution with exponent . The results showed here for Apache are representative of the distributions found in other collaborative projects.</p>", "links"=>[], "tags"=>["Network Analysis", "social networks", "software engineering", "software development", "physics", "Condensed matter physics", "Sociology", "distributions", "commits", "apache", "server", "complementary", "cumulative", "contributions", "5-day", "bins", "daughters", "branching", "exponent"], "article_id"=>1124799, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Didier Sornette", "Thomas Maillart", "Giacomo Ghezzi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103023.g005", "stats"=>{"downloads"=>1, "page_views"=>24, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Typical_distributions_of_1_st_generation_daughter_events_and_total_number_of_commits_per_developer_for_the_Apache_Web_Server_project_blue_squares_Complementary_cumulative_distribution_of_contributions_number_of_commits_per_developer_and_per_5_day_time_bi/1124799", "title"=>"Typical distributions of 1<sup>st</sup> generation daughter events and total number of commits per developer for the Apache Web Server project: (blue squares) Complementary cumulative distribution of contributions (number of commits) per developer and per 5-day time bins (1<sup>st</sup> generation daughters events in the language of the epidemic branching process described in the text) with exponent .", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 03:22:31"}

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Relative Metric

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