  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.

Physics and Society;Data Analysis, Statistics and Probability