Fractal fluctuations at mixed-order transitions in interdependent networks

Abstract

We study the geometrical features of the order parameter's fluctuations near the critical point of mixed-order phase transitions in randomly interdependent spatial networks. In contrast to continuous transitions, where the structure of the order parameter at criticality is fractal, in mixed-order transitions the structure of the order parameter is known to be compact. Remarkably, we find that although being compact, the fluctuations of the order parameter close to mixed-order transitions are fractal up to a well-defined correlation length ${\xi }^{\prime }$, which diverges when approaching the critical threshold. We characterize the self-similar nature of these critical fluctuations through their fractal dimension, $d_f^{\prime }=3d/4$, and correlation length exponent, ${\nu }^{\prime }=2/d$, where $d$ is the dimension of the system. By means of percolation and magnetization, we demonstrate that $d_f^{\prime }$ and ${\nu }^{\prime }$ are independent on the symmetry of the underlying process for any $d$ of the underlying networks.

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Cited by in corpus (4)

• Critical behavior of cascading failures in overloaded networks
• Explosive percolation in finite dimensions
• Pattern Description of Quantum Phase Transitions in the Transverse Antiferromagnetic Ising Model with a Longitudinal Field
• Information Bounds on phase transitions in disordered systems