Critical P\'olya urn

We propose a variant model of P{\'o}lya urn process, where the dynamics consist of two competing elements namely, suppression of growth and enhancement of dormant character. Here the level of such features are controlled by an internal parameter in a random manner, so as 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 that such an underlying mechanism has earlier been noted to operate in a neural network.