Scale-free networks with tunable degree-distribution exponents

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.