The gold-rush effect: how innovation speeds up

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.

Figures (7)

• Figure 1: Collective acceleration vs. individual stability in scientific and technological production. We track yearly event count, $\delta t(\tau)$ (events per year), number of active explorers, $w_{\mathrm{active}}(\tau)$, and per-capita productivity $p(\tau)=\delta t(\tau)/w_{\mathrm{active}}(\tau)$ for patents from the European Patent Office (EPO) (a) and scientific publications from the OpenAlex (OA) database (b), where natural time $\tau$ is measured in years. Each panel presents per-capita productivity (blue curve) on the left vertical axis (linear scale), and yearly event count (red curve) together with active explorers (yellow curve) on the right vertical axis (logarithmic scale). In both domains, $\delta t(\tau)$ and $w_{\mathrm{active}}(\tau)$ grow approximately exponentially, indicating strong collective acceleration. In contrast, per-capita productivity remains essentially constant. Thus, macro-level acceleration arises from rapid growth in the explorer population, not from faster individual discovery, demonstrating that distinct temporal scales govern discovery and innovation.
• Figure 2: Schematic illustration of the microscopic and collective components of the model.(a--b) Microscopic UMT dynamics. At each intrinsic-time step, an explorer extracts an element from the urn. (a) If the color has appeared before, it is reinforced by returning it with $\rho$ copies. (b) If it is new, the adjacent possible expands: $\rho$ reinforcement copies and $\nu+1$ brand-new colors are added to the urn. The extracted element is appended to the collective exploration sequence $S$. (c--d) Collective dynamics.(c) Each novelty triggers the arrival of additional explorers following the branching rule $\delta w = f(D)$. (d) In natural time $\tau$, events occur at a rate proportional to the number of active explorers, $\delta t(\tau) = a\, w(\tau)$; at each step, an explorer is sampled uniformly with replacement and follows the UMT rules. Together, these mechanisms couple microscopic exploration with population growth, providing the minimal ingredients for macroscopic innovation accelerations.
• Figure 3: Model--data comparison in natural and intrinsic time.Top panels (a--b): empirical data (markers) and model simulations (solid lines) of the cumulative number of events $t(\tau)$, the number of novelties $D(\tau)$, and the cumulative number of distinct explorers $D_w(\tau)$ as functions of natural time $\tau$ for patents (EPO) (a) and scientific publications (OA) (b). Here $t(\tau)$ counts all patents or publications produced, $D(\tau)$ counts novelties defined as first appearances of new technological or scientific combinations (IPC-code combinations or keyword combinations), and $D_w(\tau)$ counts the cumulative number of distinct inventors/authors observed up to time $\tau$. Calibrated through the branching and productivity relations, the model reproduces the empirical growth of all three quantities and captures the acceleration induced by the expanding explorer population. Bottom panels (c--d): the same quantities expressed in intrinsic time $t$, which counts cumulative discovery events. In this representation, both novelties and explorer introductions follow approximately linear microscopic trends, showing that the coupled urn--branching dynamics provides a consistent description across temporal scales.
• Figure 4: Frequency distribution of technological and scientific elements. Empirical data (markers) and model simulations (solid red lines) for patents (EPO) (a) and scientific publications (OA) (b). The dashed black line indicates the reference scaling $p(f)\sim f^{-2}$. The horizontal axis reports the frequency $f$ of each distinct element (an IPC combination or a scientific keyword), and the vertical axis shows the corresponding probability density $p(f)$ on log--log scales. Both domains display Zipf-like statistics: the probability density follows approximately $p(f)\sim f^{-2}$, corresponding to a Zipf rank--frequency exponent close to one, a common signature of discovery and innovation processes. The model reproduces this heavy-tailed distribution when run with the microscopic urn dynamics in the regime $\nu=\rho=1$, showing that the same minimal mechanism that matches temporal trends also captures the statistical structure of the explored elements.
• Figure 5: Model behavior across branching regimes in intrinsic and natural time. The model is simulated with urn parameters $\nu=\rho=1$, initial condition $N_0=1$, and $a=1$, $w_0=0$. Panels(a–c) show quantities in intrinsic time $t$, while panels (d–f) display the corresponding dynamics in natural time $\tau$, obtained from $dt/d\tau = a w(t(\tau))$. (a) The number of distinct elements follows the UMT prediction $D(t)\sim t/\log t$, identical across branching regimes. (b) Growth of the explorer population $w(t)$ for logarithmic ($w\sim \log t$), linear ($w\sim t$), and exponential ($w\sim e^{t}$) branching processes. (c) Frequency–rank distributions on log--log scales, displaying the Zipf-like behavior $f(R)\sim R^{-1}$ generated by the microscopic urn dynamics. (d–f) Dynamics in natural time $\tau$ for the three branching cases. (d): Logarithmic branching leads to slow, near–intrinsic behavior; (e): linear branching produces exponential growth; (f): exponential branching generates a finite-time singularity, with $D(\tau)$ and $w(\tau)$ diverging super-exponentially. These regimes illustrate how different branching forms lead to distinct macroscopic innovation patterns despite identical microscopic exploration patterns.