How Much is the Whole Really More than the Sum of its Parts? 1 + 1 = 2.5: Superlinear Productivity in Collective Group Actions

TL;DR

This paper investigates superlinear productivity in open source projects, revealing that doubling group size can increase output by about 2.5 times, and models this phenomenon through criticality and cascade mechanisms.

Contribution

It introduces a unifying cascade model explaining superlinear productivity and provides the first empirical test linking event size distributions to critical branching processes.

Findings

Superlinear growth with exponent ~4/3 across groups of 5 to hundreds.

Doubling group size increases output by approximately 2.5 times.

Smaller groups show more variability and higher superlinearity.

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.