How Much Is the Whole Really More than the Sum of Its Parts? 1 ⊞ 1 = 2.5: Superlinear Productivity in Collective Group Actions

In a variety of open source software projects, we document a superlinear growth of production intensity ([Image: see text]) as a function of the number of active developers [Image: see text], with a median value of the exponent [Image: see text], with large dispersions of [Image: see text] from slightly less than [Image: see text] up to [Image: see text]. For a typical project in this class, doubling of the group size multiplies typically the output by a factor [Image: see text], 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.