  The relationships between diversity, productivity and scale determine much of
the structure and robustness of complex biological and social systems. While
arguments for the link between specialization and productivity are common,
diversity has often been invoked as a hedging strategy, allowing systems to
evolve in response to environmental change. Despite their general appeal, these
arguments have not typically produced quantitative predictions for optimal
levels of functional diversity consistent with observations. One important
reason why these relationships have resisted formalization is the idiosyncratic
nature of diversity measures, which depend on given classification schemes.
Here, we address these issues by analyzing the statistics of professions in
cities and show how their probability distribution takes a universal
scale-invariant form, common to all cities, obtained in the limit of infinite
resolution of given taxonomies. We propose a model that generates the form and
parameters of this distribution via the introduction of new occupations at a
rate leading to individual specialization subject to the preservation of access
to overall function via their ego social networks. This perspective unifies
ideas about the importance of network structure in ecology and of innovation as
a recombinatory process with economic concepts of productivity gains obtained
through the division and coordination of labor, stimulated by scale.
