Critical Pólya urn

We propose a variant model of the Pólya urn process, where the dynamics consist of two competing elements: suppression of growth and enhancement of dormant character. Here the level of such features is controlled by an internal parameter in a random manner, so that the system self-organizes and characteristic observables exhibit scale invariance, suggesting the existence of criticality. Varying the internal control parameter, one can explain interesting universality classes for avalanche-type events. We also discuss different versions of the model. It is pointed out that such an underlying mechanism has earlier been noted to operate in a neural network.

Figure 1

Sketch of the model dynamical rule consisting of two steps: (i) A ball is randomly drawn and its color is checked. (ii) The ball is returned with one more ball of the same or of a different color, probabilistically [see Eqs. (2) and (3)]. There are black and white balls in the urn.

Figure 2

Top panel: A typical trajectory for the critical Pólya urn process depicted by a staircase function for parameter $\lambda =1$ as a 2D random walk. Here the walk starts from {2, 1} and stops at {13, 13}, thus $L=13$ with $T=23$. Bottom panel: Corresponding 1D random walk path.

Figure 3

The distribution $P\left(L\right)$ for different values of the parameter $\lambda$. We take ${10}^8$ independent realizations, and the maximum run time for a process is ${10}^4$. Log-binned data are presented. For comparison, two straight lines are drawn with slopes 3/2 (upper line) and 5/2 (lower line).

Figure 4

The variation of numerically estimated exponent ${\tau }_L$ with mixing parameter $z\equiv \lambda ,\mu ,\delta$ for critical Pólay urn processes: I, II, III, respectively. Here $1.25\le {\tau }_L\le 2.5$.

Figure 5

Top panel: The cluster size distribution $P\left(S\right)$ for $a=1$ and different values of parameter $\lambda$, along with the best fit curves shown by dashed lines. The slopes are ${\tau }_S=$ 2, 5/3, 4/3 for $\lambda =1$, 0.5, and 0, respectively. Bottom panel: The plot of $P\left(S\right)$ for parameters $a=2$ and $\lambda =1/2$. A straight line with slope 3/2 is drawn for comparison.

Figure 6

Signature of self-organization: A typical plot of the scaled indicator function $\sqrt{t}[f_{\lambda }\left(t\right)-0.5]$.

Figure 7

Data collapse curves for the distribution $P\left(L\right)$ with different values of $c$ and $\lambda$. Here $\nu =(L-\langle L\rangle )/{\sigma }_L$ and $J\left(\nu \right)=\sqrt{2\pi {\sigma }_L^2}P\left(L\right)$.