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 interevent time distributions, and nontrivial 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.
Highlights
Known as Zipf’s law or Pareto distributions, power-laws and long-tailed distributions are scattered among many natural systems[1]
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
In this work we have studied and characterized a preferential growth stochastic model with a bounded memory kernel
Summary
Known as Zipf’s law or Pareto distributions, power-laws and long-tailed distributions are scattered among many natural systems[1]. The popularity distribution of the words generated by this process is a skewed distribution with a power-law tail of exponent α = 2 + p/(1 − p), meaning that the number ns of words with popularity s ≫ 1 satisfy ns ∼ s−α Another interesting effect that is frequently observed in empirical systems that display a long-tailed frequency distribution, is the presence of memory in the form of long-range correlations[13,14,15,16] or a bursty dynamics[17,18]. In a recent work[22] we have shown that the long-range correlations introduced by Cattuto’s model allow to describe the memory effects observed in the growing process of an extensive chess database This model fails to describe the inter-event time distribution in the use of popular game lines by individual players.
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