Enhancing resilience of interdependent networks by healing

Interdependent networks are characterized by two kinds of interactions: The usual connectivity links within each network and the dependency links coupling nodes of different networks. Due to the latter links such networks are known to suffer from cascading failures and catastrophic breakdowns. When modeling these phenomena, usually one assumes that a fraction of nodes gets damaged in one of the networks, which is followed possibly by a cascade of failures. In real life the initiating failures do not occur at once and effort is made to replace the ties eliminated due to the failing nodes. Here we study a dynamic extension of the model of interdependent networks and introduce the possibility of link formation with a probability w, called healing, to bridge non-functioning nodes and enhance network resilience. A single random node is removed, which may initiate an avalanche. After each removal step healing starts resulting in a new topology. Then a new node fails and the process continues until the giant component disappears either in a catastrophic breakdown or in a smooth transition. Simulation results are presented for square lattices as starting networks under random attacks of constant intensity. We find that the shift in the position of the breakdown has a power-law scaling as a function of the healing probability with an exponent close to  1. Below a critical healing probability, catastrophic cascades form and the average degree of surviving nodes decreases monotonically, while above this value there are no macroscopic cascades and the average degree has first an increasing character and decreases only at the very late stage of the process. These findings facilitate to plan intervention in case of crisis situation by describing the efficiency of healing efforts needed to suppress cascading failures.