An Explanation of Universality in Growth Fluctuations

Abstract

Phenomena as diverse as breeding bird populations, the size of U.S. firms, money invested in mutual funds, and the scientific output of universities all show unusual but remarkably similar growth fluctuations. The fluctuations display characteristic features, including heavy tails and anomalous power law scaling of the standard deviation as a function of size. Many theories have now been put forward to explain this, all of them based on modifications and extensions of proportional growth of subunits. We analyze data from bird populations, firms, and mutual funds and show that the growth fluctuations match a Levy distribution very well. This was previously suggested by Wyart and Bouchaud and Gabaix, but until now never tested. However, we show that their theory (and indeed all previous theories) are ruled out, at least for these three data sets, because they require size distributions that are too heavy tailed. We introduce a simple additive replication model, in which groups (such as firms) grow by replacing each of their members by a random number of new members. To demonstrate how the individual growth fluctuations can be heavy-tailed even though the sizes are not, we propose a model based on stochastic influence dynamics over a scale-free contact network, and show that it produces the correct behavior. We generalize the model to the case where some groups are preferred over others, and show that this can lead to a breakdown of the anomalous scaling, which appears to be observed for some other data sets.