History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space-reducing (SSR) processes necessarily lead to Zipf’s law in the rank distributions of their outcomes.

We show that by adding noise to SSR processes the corresponding rank distributions remain exact power laws, p(x)∼x−λ, where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from α=2 to ∞.

We discuss several applications showing how SSR processes can be used to understand Zipf’s law in word frequencies, and how they are related to diffusion processes in directed networks, or aging processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organized critical processes.

B. Corominas-Murtra, R. Hanel, S. Thurner, Understanding scaling through history-dependent processes with collapsing sample space, PNAS 112 (2015) 5348–5353