Power-law distributions from additive preferential redistributions

We introduce a nongrowth model that generates the power-law distribution with the Zipf exponent. There are $N$ elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule, while the sum of quantities is conserved. The situation described by this model is similar to those of closed $N$-particle systems when conservative two-body collisions are only allowed. We obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model, and find that the model exhibits the scaling behavior for some parameter ranges. Unlike well-known growth models, this alternative mechanism generates the power-law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements. This model can be applied to some examples such as personal wealths, city sizes, and the generation of scale-free networks when only rewiring is allowed.

Figure 1

Evolution of the cumulative distribution $P(⩾k)$ when $N={10}^5$, $\alpha =1$, and $\beta =0.1$ in a log-log plot. From an initial distribution $P\left(k\right)={\delta }_{k1}$, we observe how $P(⩾k)$ evolves as the number of timesteps $T$ varies from 0 to ${10}^9$ (엯), ${10}^{10}$ (◻), ${10}^{11}$ (◇), ${10}^{12}$ (▵). We can observe that $P(⩾1)\simeq \beta (=0.1)$, which leads us to $P\left(0\right)=1-P(⩾1)\simeq 1-\beta$. The dashed line represents the theoretical stationary distribution $P(⩾k)=1∕(9k+1)$ at $\beta =0.1$.

Figure 2

When $N={10}^4$ and $T={10}^9–{10}^{10}$, $P\left(0\right)$ values are found numerically for various $\alpha$ and $\beta$ values (averaged over ten runs). (a) $P\left(0\right)$ versus $\alpha$ when $\beta =0.1$ (엯), 0.3 (◻), 0.5 (◇), 0.7 (▵), 0.9 (∗). $P\left(0\right)$ is close to $1-\beta$ when $\alpha$ is greater than a certain value for a given $\beta$. (b) $P\left(0\right)$ when $\beta$ for $\alpha =1$ (엯), 5 (◻), 10 (◇), 100 (▵). $P\left(0\right)$ is close to $1-\beta$ when $\beta$ is less than a certain value for a given $\alpha$. Moreover, we observe that $P\left(0\right)=1∕(1+\alpha )$ from Eq. (7) are satisfied when $\beta =1$. The dashed line represents $P\left(0\right)=1-\beta$.

Figure 3

The scaling condition in $(\alpha ,\beta )$ space. The dashed line represents $f(\alpha ,\beta )=ϵ$ when $ϵ={10}^{-3}$. As $\alpha$ increases, the range of $\beta$ for the power law approaches $0<\beta <1$.

Figure 4

Stationary distributions for various $(\alpha ,\beta )$ values. Dashed lines represent theoretical stationary probability distributions for given $\beta$ values, and data points are logarithmically binned for scaling cases. (a) When $N={10}^6$ and $T=7\times {10}^{11}$, $\alpha$ is fixed at 5, and $\beta =0.2$ (엯), 0.4 (◻), 0.6 (◇), 0.8 (▵), 1.0 (∗). The power-law distributions are observed clearly when $\beta =0.2$ and 0.4. (b) Examples of various $(\alpha ,\beta )$ values exhibiting power-law distributions after $T={10}^{12}$: $N={10}^6$, $\alpha =1$, $\beta =0.2$ (엯); $N={10}^5$, $\alpha =10$, $\beta =0.4$ (◻); $N={10}^4$, $\alpha =100$, $\beta =0.6$ (◇); $N={10}^3$, $\alpha =1000$, $\beta =0.8$ (▵).