A short-ranged memory model with preferential growth

TL;DR

This paper introduces a finite-memory variant of the Yule-Simon model that captures empirical power-law distributions and phase transition phenomena in complex systems, highlighting the role of bounded memory and innovation rates.

Contribution

It develops a new preferential growth model with bounded memory, analyzes its properties, and demonstrates phase transition behavior and realistic temporal dynamics.

Findings

Distributions follow power-laws with empirical exponents.

Model exhibits a phase transition at critical innovation rate.

Time series show bursty, solar flare-like activity.

Abstract

In this work we introduce a variant of the Yule-Simon model for preferential growth by incorporating a finite kernel to model the effects of bounded memory. We characterize the properties of the model combining analytical arguments with extensive numerical simulations. In particular, we analyze the lifetime and popularity distributions by mapping the model dynamics to corresponding Markov chains and branching processes, respectively. These distributions follow power-laws with well defined exponents that are within the range of the empirical data reported in ecologies. Interestingly, by varying the innovation rate, this simple out-of-equilibrium model exhibits many of the characteristics of a continuous phase transition and, around the critical point, it generates time series with power-law popularity, lifetime and inter-event time distributions, and non-trivial temporal correlations, such as a bursty dynamics in analogy with the activity of solar flares. Our results suggest that an appropriate balance between innovation and oblivion rates could provide an explanatory framework for many of the properties commonly observed in many complex systems.