Short-ranged memory model with preferential growth

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.

Figure 1

Outline of BMPG model process depicting how an element is incorporated to the time series at time $t$. With probability $p$ the added element belongs to a new class, i.e., a new color ball is introduced. With complementary probability $1-p$ an existing element is copied, chosen uniformly among the last elements of the series. Elements that are more than $\kappa$ time steps away—the shaded balls in the figure—cannot be copied.

Figure 2

Lifetime distribution calculated for the BMPG model using $\kappa =500$ for $p=p_c=1/\kappa$, $p=0.01p_c$, and $p=10p_c$. The lines correspond to power-laws with different exponents.

Figure 3

Top panel: Popularity distribution calculated for the generated sequences with BMPG model for different kernel sizes. The straight lines correspond to fits of Eq. (7). In all cases the fitted exponents are $\alpha \simeq 3/2$. Bottom panel: Popularity distribution calculated for the generated sequences with the BMPG model for $\kappa =100$ and $p=0.1p_c$, $p_c$, and $10p_c$; the full lines correspond to the estimation in Eq. (11).

Figure 4

Autocorrelation function $C\left(\mathrm{\Delta }t\right)$ calculated for sequences generated with the BMPG model for different kernel sizes and $p=p_c$, rescaled by ${\kappa }^{1.9}$. Inset: Decay characteristic time $R$ of the autocorrelation function as a function of the memory kernel extension $\kappa$. The straight line corresponds to a linear function fit in log-log with exponent $\gamma =1.9$.

Figure 5

Top panel: Distribution of interevent times for the BMPG model for with $\kappa =500$ and three values of $p$. We also show a fit according to Eq. (16) (black full line) for the case of $p=p_c$. For comparison, we show the distribution of interevent times for a sequence generated with CLS model for ${\kappa }_C=500$ and $p=1/{\kappa }_C$ and a fit with Eq. (15) (gray dashed line). Middle panel: Distribution of interevent times for different levels of activity, rescaled by ${\langle s\rangle }^{-1/2}$ and a power law with exponent 3 for comparison. Bottom panel: Burstiness parameter $B$ calculated for the BMPG model for various values of $\kappa$ and $p$ as a function of $p\kappa$; the black vertical line corresponds to $p\kappa =1$ and the horizontal lines, to the values of $B$ calculated from the distribution $f\left(\tau \right)$.

Figure 6

Order parameter mean $\langle \varphi \rangle$ (top panel) and fluctuations $(\langle {\varphi }^2\rangle -{\langle \varphi \rangle }^2)\kappa [-ln\left(p\right)]$ (bottom panel) calculated for the BMPG model for various values of $\kappa$ as a function of $p$ and rescaled by $p_c=1/\kappa$ (Inset).

Figure 7

Top panel: kernel configurations for the supercritical ($p>p_c$), critical ($p=p_c$), and subcritical ($p<p_c$) states. Bottom panel: Order parameter distribution calculated for the BMPG model for various values of $\kappa$ in the case of $p=p_c$. Inset: Examples of the order parameter distribution in the case of $p>p_c$ and $p<p_c$.

Figure 8

Entropy mean ${\mu }_S$ (top panel) and fluctuations ${\sigma }_S$ (bottom panel) calculated for the BMPG model for various values of $\kappa$ as a function of $p\kappa$.