The gold-rush effect: how innovation speeds up

TL;DR

This paper introduces a multi-scale model explaining how collective innovation accelerates rapidly despite individual discovery constraints, by linking population growth of explorers to macro-level exponential growth.

Contribution

It presents a novel minimal model combining the Theory of the Adjacent Possible and Urn Model with Triggering to explain macro acceleration from micro-level discovery processes.

Findings

Model accurately reproduces exponential growth in patent and publication data.

Explains the nonlinear relationship between individual discoveries and collective innovation.

Provides a unified framework linking micro-level behavior to macro-level innovation dynamics.

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.