Universality and scaling laws in the interdependent network model with healing

Cascading failures may lead to dramatic collapse in interdependent networks, where the breakdown takes place as a discontinuity of the order parameter. However, this is a hybrid transition, meaning that, besides this first order character, the transition shows scaling too. Recently we showed that there are two sets of exponents describing respectively the order parameter and the cascade statistics, which are connected by a scaling law. Here we study the question of universality of these exponents in models controlled by a parameter, which at some value suppresses the discontinuity of the order parameter. In interdependent networks with healing there are two universality classes: Below the critical healing value the exponents agree with those of the original model, while above this value the model displays trivial scaling. If the range r of dependency links is smaller than a critical value rmax the transition is of second order and results on the exponents indicate universal behavior. As rmax is approached finite size effects hinder the accurate numerical determination of the exponents. For r > rmax the relation between the range of dependency and the system size N determines the behavior. For rmax < r √ N there seems to be universal behavior different from that of the infinite range case while in the opposite limit we observe exponents of the original model.