Hybrid Phase Transition into an Absorbing State: Percolation and Avalanches

TL;DR

This paper investigates hybrid phase transitions in interdependent networks, revealing critical phenomena such as divergence of fluctuations and power-law avalanche distributions at the transition point, with specific critical exponents.

Contribution

It provides a detailed analysis of the critical behaviors and exponents associated with hybrid percolation transitions in interdependent networks, including avalanche dynamics.

Findings

Critical fluctuations of the order parameter diverge at transition.

Finite avalanches follow a power-law size distribution.

Order parameter exponent $eta_m$ is 1/2, with system-dependent $oldsymbol{ extgamma}_m$.

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\H{o}s--R\'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 $\beta_m$ of the order parameter is $1/2$ under general conditions, while the value of the exponent $\gamma_m$ 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, $\beta_a$ and $\gamma_a$. These two critical behaviors are coupled by a scaling law: $1-\beta_m=\gamma_a$.