Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks

We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ζ′ that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension df′=3d/4, and correlation length exponent ν′=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that df′ and ν′ are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.