  Phenomena as diverse as breeding bird populations, the size of U.S. firms,
money invested in mutual funds, the GDP of individual countries and the
scientific output of universities all show unusual but remarkably similar
growth fluctuations. The fluctuations display characteristic features,
including double exponential scaling in the body of the distribution and power
law scaling of the standard deviation as a function of size. To explain this we
propose a remarkably simple additive replication model: At each step each
individual is replaced by a new number of individuals drawn from the same
replication distribution. If the replication distribution is sufficiently heavy
tailed then the growth fluctuations are Levy distributed. We analyze the data
from bird populations, firms, and mutual funds and show that our predictions
match the data well, in several respects: Our theory results in a much better
collapse of the individual distributions onto a single curve and also correctly
predicts the scaling of the standard deviation with size. To illustrate how
this can emerge from a collective microscopic dynamics we propose a model based
on stochastic influence dynamics over a scale-free contact network and show
that it produces results similar to those observed. We also extend the model to
deal with correlations between individual elements. Our main conclusion is that
the universality of growth fluctuations is driven by the additivity of growth
processes and the action of the generalized central limit theorem.
