Time-Dependent Urn Models reproduce the full spectrum of novelties discovery

TL;DR

This paper introduces the Time-dependent Urn Model with Triggering (TUMT), a flexible framework that captures diverse innovation patterns in non-stationary systems, unifying phenomena like Heaps' and Zipf's laws.

Contribution

The TUMT generalizes previous urn models by incorporating time-dependent parameters, enabling it to model a wider range of empirical innovation dynamics.

Findings

TUMT reproduces various observed innovation behaviors.

It identifies a critical region where Heaps' and Zipf's laws coexist.

Exponents for the coexisting laws are analytically computed.

Abstract

Systems driven by innovation, a pivotal force in human society, present various intriguing statistical regularities, from the Heaps' law to logarithmic scaling or somewhat different patterns for the innovation rates. The Urn Model with Triggering (UMT) has been instrumental in modelling these innovation dynamics. Yet, a generalisation is needed to capture the richer empirical phenomenology. Here, we introduce a Time-dependent Urn Model with Triggering (TUMT), a generalisation of the UMT that crucially integrates time-dependent parameters for reinforcement and triggering to offer a broader framework for modelling innovation in non-stationary systems. Through analytical computation and numerical simulations, we show that the TUMT reconciles various behaviours observed in a broad spectrum of systems, from patenting activity to the analysis of gene mutations. We highlight how the TUMT features a "critical" region where both Heaps' and Zipf's laws coexist, for which we compute the exponents.