@article{lee2016hybrid,
author = "Deokjae Lee and S. Choi and M. Stippinger and J. Kert{\'e}sz and B. Kahng",
abstract = "Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it. Here we study these phenomena on the Erd$\backslash$H{\{}o{\}}s--R$\backslash$'enyi and the two dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors: divergence of the fluctuations of the order parameter and power-law size distribution of finite avalanches at a transition point. At the transition point, avalanches of infinite size occur thus the avalanche statistics also has the nature of a HPT. The exponent {\$}\backslashbeta{\_}m{\$} of the order parameter is {\$}1/2{\$} under general conditions, while the value of the exponent {\$}\backslashgamma{\_}m{\$} characterizing the fluctuations of the order parameter depends on the system. The critical behavior of the finite avalanches can be described by another set of exponents, {\$}\backslashbeta{\_}a{\$} and {\$}\backslashgamma{\_}a{\$}. These two critical behaviors are coupled by a scaling law: {\$}1-\backslashbeta{\_}m=\backslashgamma{\_}a{\$}.",
doi = "10.1103/PhysRevE.93.042109",
issn = "15502376",
journal = "Physical Review E - Statistical, Nonlinear, and Soft Matter Physics",
number = "4",
pages = "1--11",
title = "{H}ybrid phase transition into an absorbing state: {P}ercolation and avalanches",
volume = "93",
year = "2016",
}