Abstract
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, interaction-based and 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, 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.
Highlights
Since at least Aristotle, the adage in the title has permeated human thinking, with prominent influence in psychology (Gestalt theory [1]), biology, physics, economics [6,7] among a wealth of other examples
Prominent among other developments are the fields of complexity science, synergetics and complex adaptive system theory, which strive to understand natural and social systems in terms of a systemic or holistic approach, where the above adage is translated into the scientific concept of emergence that results from repetitive interactions between simple constituting elements in extended out-of-equilibrium adaptive systems
Expanding on the remark on the different sizes involved in our open source software (OSS) database compared with cities, we present a simple mechanism and theoretical argument that may explain the smaller value of the superlinear exponent for cities, deriving it from our results obtained for small group sizes
Summary
Since at least Aristotle, the adage in the title has permeated human thinking, with prominent influence in psychology (Gestalt theory [1]), biology (brain functions [2], ecological networks [3]), physics (spontaneous symmetry breaking [4] and the ‘‘more is different’’ concept [5]), economics [6,7] among a wealth of other examples. Within the epidemic framework presented, Ptwot(r) will be shown to be equivalent to the statistics of the cluster sizes of contributions following critical cascades [36] (see expression (12)), i.e., when the dynamics of triggering of activity is close to or at the critical point of a branching process This result, showed for the Apache Web Server project, is representative of the distributions found in other collaborative projects. According to equation (8), the typical total production (number of commits) by c developers is proportional to c1=m, when their contributions are wildly distributed with a power law distribution with exponent mv1 According to this large deviation mechanism, the superlinear exponent b is equal to 1=m. A similar story is likely to be relevant in large OSS projects, groups and firms, which for a variety of reasons ranging from cognitive limitations [61] to efficiency maximization [62] are found to organize in subgroups, often in a hierarchical way [61]
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