Weakest-Link Scaling and Extreme Events in Finite-Sized Systems

Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the κ-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the κ-Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution.