  Zipf's power law is a general empirical regularity found in many natural and
social systems. 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 report a detailed empirical analysis of a burgeoning network of
social groups, in which all ingredients needed for Zipf's law to apply are
verifiable and verified. We estimate empirically the average growth $r$ and its
standard deviation $\sigma$ as well as the death rate $h$ and predict without
adjustable parameters the exponent $\mu$ of the power law distribution $P(s)$
of the group sizes $s$. The predicted value $\mu = 0.75 \pm 0.05$ is in
excellent agreement with maximum likelihood estimations. According to theory,
the deviation of $P(s)$ from Zipf's law (i.e., $\mu < 1$) constitutes a direct
statistical quantitative signature of the overall non-stationary growth of the
social universe.
