Statistical resilience of random populations to random perturbations

We consider populations represented by random collections of real-valued points, and explore their statistical resilience to random perturbations-seeking populations whose statistics remain qualitatively unchanged by the action of arbitrary random perturbations of a certain type. Studying a general physical perturbation scheme, we obtain an explicit characterization of statistically resilient populations, show that these objects are fractal, and comprehensively analyze their topological and statistical structures. An application of statistical resilience attained is an alternative explanation of the ubiquity of power-law statistics.