Full spectrum of novelties in time-dependent urn models

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 modeling these innovation dynamics. Yet a generalization is needed to capture the richer empirical phenomenology. Here we introduce a time-dependent urn model with triggering (TUMT), a generalization of the UMT that crucially integrates time-dependent parameters for reinforcement and triggering to offer a broader framework for modeling innovation in nonstationary systems. Through analytical computation and numerical simulations, we show that the TUMT reconciles various behaviors 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.