The gold-rush effect: how innovation speeds up

Author

Bellina, Alessandro and Di Bona, Gabriele and De Marzo, Giordano and Loreto, Vittorio

Abstract

Innovation records often exhibit ``hockey-stick'' patterns of abrupt, near-singular growth at the collective level. However, this macroscopic explosiveness stands in stark contrast to individual discovery, which remains bounded by cognitive and temporal constraints and follows slow, sublinear accumulation laws. Here, we resolve this micro-macro discrepancy by introducing a minimal multi-scale model that identifies the growth of the explorer population as the primary driver of aggregate acceleration. Building on the Theory of the Adjacent Possible and the Urn Model with Triggering (UMT), we demonstrate that as discoveries expand the space of possibilities, they attract new explorers through a self-reinforcing branching process. This expansion induces a nonlinear mapping between intrinsic time (individual discovery events) and natural time (calendar years), effectively reparameterizing steady individual trajectories into accelerating system-level dynamics. We validate the framework using large-scale patent (EPO) and scientific publication (OpenAlex) datasets, showing that the model accurately reproduces stable per-capita productivity alongside exponential aggregate growth. By providing a quantitative link between individual behavior and collective takeoffs, this work offers a unified foundation for understanding the statistical structure and temporal evolution of innovation ecosystems.