Universality and scaling laws in the cascading failure 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. In the cascading failure (CF) model with healing there is a control parameter which at some value suppresses the discontinuity of the order parameter. However, up to this value of the healing parameter the breakdown is a hybrid transition, meaning that, besides this first order character, the transition shows scaling too. In this paper we investigate the question of universality related to the scaling behavior. Recently we showed that the hybrid phase transition in the original CF model has two sets of exponents describing respectively the order parameter and the cascade statistics, which are connected by a scaling law. In the CF model with healing we measure these exponents as a function of the healing parameter. We find two universality classes: In the wide range below the critical healing value the exponents agree with those of the original model, while above this value the model displays trivial scaling meaning that fluctuations follow the central limit theorem.

[1]  János Kertész,et al.  Enhancing resilience of interdependent networks by healing , 2013, ArXiv.

[2]  Peter Grassberger,et al.  Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  H. Stanley,et al.  Networks formed from interdependent networks , 2011, Nature Physics.

[4]  Wei Li,et al.  Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links , 2012, Physical review letters.

[5]  S. Havlin,et al.  Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Mikkel Thorup,et al.  Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity , 2001, JACM.

[7]  S. Havlin,et al.  Simultaneous first- and second-order percolation transitions in interdependent networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Deokjae Lee,et al.  Hybrid phase transition into an absorbing state: Percolation and avalanches. , 2015, Physical review. E.

[9]  Lidia A. Braunstein,et al.  Multiple tipping points and optimal repairing in interacting networks , 2015, Nature Communications.

[10]  H E Stanley,et al.  Recovery of Interdependent Networks , 2015, Scientific Reports.

[11]  H. Stanley,et al.  Percolation of partially interdependent scale-free networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Harry Eugene Stanley,et al.  Catastrophic cascade of failures in interdependent networks , 2009, Nature.

[13]  Deokjae Lee,et al.  Efficient algorithm to compute mutually connected components in interdependent networks. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..