title: Hybrid phase transition into an absorbing state: Percolation and avalanches.
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 Erdős-Rényi 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 global or "infinite" avalanches occur, while the finite ones have a power law size distribution; thus the avalanche statistics also has the nature of a HPT. The exponent β_{m} of the order parameter is 1/2 under general conditions, while the value of the exponent γ_{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, β_{a} and γ_{a}. These two critical behaviors are coupled by a scaling law: 1-β_{m}=γ_{a}.
authors.affiliation: CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea.
authors.forename: Deokjae
authors.initials: D
authors.lastname: Lee
authors.affiliation: CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea.
authors.forename: S
authors.initials: S
authors.lastname: Choi
authors.affiliation: Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest H-1111, Hungary.
authors.forename: M
authors.initials: M
authors.lastname: Stippinger
authors.affiliation: Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest H-1111, Hungary.|Center for Network Science, Central European University, Budapest H-1051, Hungary.
authors.forename: J
authors.initials: J
authors.lastname: Kertész
authors.affiliation: CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea.
authors.forename: B
authors.initials: B
authors.lastname: Kahng