  Probability distributions having power-law tails are observed in a broad
range of social, economic, and biological systems. We describe here a
potentially useful common framework. We derive distribution functions $\{p_k\}$
for situations in which a `joiner particle' $k$ pays some form of price to
enter a `community' of size $k-1$, where costs are subject to
economies-of-scale (EOS). Maximizing the Boltzmann-Gibbs-Shannon entropy
subject to this energy-like constraint predicts a distribution having a
power-law tail; it reduces to the Boltzmann distribution in the absence of EOS.
We show that the predicted function gives excellent fits to 13 different
distribution functions, ranging from friendship links in social networks, to
protein-protein interactions, to the severity of terrorist attacks. This
approach may give useful insights into when to expect power-law distributions
in the natural and social sciences.
