Universalities in the Avalanche Dynamics of Novelties and Non-Novelties and some Notes on the Heaps law

Abstract

Unprecedented events intertwine with the repetition of the past in natural phenomena and human activities. Key statistical patterns, such as Heaps' and Taylor's laws and Zipf's law, have been identified as characterizing the dynamical processes that govern the emergence of novelties and the abundance of repeated elements. Observing these statistical regularities has been pivotal in motivating the search for modeling schemes that can explain them and clarify key mechanisms underlying the appearance of new elements and their subsequent recurrence. In this study, we analyze sequences of novel and non-novel elements, referred to as avalanches, in real-world systems. We show that avalanche statistics provide a complementary characterization of innovation dynamics, extending beyond the three fundamental laws mentioned above. Although arising from collective dynamics, some systems behave as a single instance of a stochastic process. Others, such as natural language, exhibit features that we can only explain by a superposition of different dynamics. This distinction is not apparent when considering Heaps' law alone, while it clearly emerges in the avalanche statistics. By interpreting these empirical observations, we also advance the theoretical understanding of urn-based models that successfully reproduce the observed behaviors associated with Heaps', Zipf's, and Taylor's laws. We derive analytical expressions that accurately describe the probability distributions of avalanches and the Heaps law beyond its asymptotic regime. Building on these results, we derive a scaling relation that we show also holds in real-world systems, indicating a form of universality in the dynamics of novelty.