The dynamics of discovery and the Heaps-Zipf relationship

Abstract

When following a sequence - such as reading a text or tracking a user's activity - one can measure how the "dictionary" of distinct elements (types) grows with the number of observations (tokens). When this growth follows a power law, it is referred to as Heaps' law, a regularity often associated with Zipf's law and frequently used to characterize human discovery processes. While random sampling from a Zipf-like distribution can reproduce Heaps' law, this connection relies on the assumption of temporal independence - an assumption often violated in real-world systems although frequently found in the literature. Here, we investigate how temporal correlations in token sequences affect the type-token curve. In human behaviors like music listening and web browsing, domain-specific correlations in token ordering lead to systematic deviations from the Zipf-Heaps framework, effectively decoupling the type-token plot from the rank-frequency distribution. Using a minimal one-parameter model, we reproduce a wide variety of type-token trajectories, including the extremal cases that bound all possible behaviors compatible with a given frequency distribution. Our results demonstrate that type-token growth reflects not only the empirical distribution of type frequencies, but also the domain-specific, temporal structure of the sequence - a factor often overlooked in empirical applications of scaling laws to characterize human behavior.