Empirical test of the origin of Zipf’s law in growing social networks

Zipf’s power law is a general empirical regularity found in many systems. We report a detailed analysis of a burgeoning network of social groups, in which all ingredients needed for Zipf’s law to apply are verifiable and verified. A recently developed theory predicts that Zipf’s law corresponds to systems that are growing according to a maximally sustainable path in the presence of random proportional growth, stochastic birth and death processes. We estimate empirically the average growth r and its standard deviation σ as well as the death rate h and predict without adjustable parameters the exponent μ of the power law distribution P(s) of the group sizes s. Using numerical simulations of the underlying growth model, we demonstrate that the empirical stability of Zipf’s law over the whole lifetime of the social network can be attributed to the interplay between a finite lifetime effect and a large σ value. Our analysis and the corresponding results demonstrate that Zipf’s law can be observed with a good precision even when the balanced growth condition is not realized, if the random proportional growth has a strong stochastic component and is acting on young systems under development.