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 ${\xi }^{\prime }$ that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension $d_f^{\prime }=3d/4$, and correlation length exponent ${\nu }^{\prime }=2/d$, where $d$ is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that $d_f^{\prime }$ and ${\nu }^{\prime }$ are the same for both, i.e., independent of the symmetry of the process for any $d$ of the underlying networks.

Figure 1

Fluctuations at a mixed-order transition. (a) Illustration of the model of randomly interdependent $d$-dimensional networks (here $d=2$) studied in this Letter, featuring short-range connectivity links (gray links) and long-range dependency links (orange couplings). (b) Each realization of a hybrid transition [see (c) for a close-up of the bounded region] has its own critical threshold $a_c$ and critical mass ${\mathcal{O}}_c\equiv \mathcal{O}(a_c)$—related to each other via the scaling law [9] $\mathcal{O}(a)\sim {\mathcal{O}}_c+|a-a_c{|}^{1/2}$—whose distributions follow a certain profile, as shown in (d) for $P(a_c)$ and in (e) for $P({\mathcal{O}}_c)$. (f) Illustration of the fluctuating values of the critical mass, their mean value $⟨{\mathcal{O}}_c⟩$, and their statistical variation ${\sigma }^2({\mathcal{O}}_c)$.

Figure 2

Correlation length exponents of fluctuations length. (a) Scaling of $\sigma (p_c)$ with $L$, for all studied dimensions ($d=2–8$). We find excellent agreement between simulations and the scaling relation in Eq. (2). Inset: the dependence of ${\nu }^{\prime }$ on the dimension $d$ of the underlying lattices follows the relation ${\nu }^{\prime }=2/d$. (b) The distribution of $p_c$ (here, $d=2$) fits a skewed Gaussian and follow the scaling in Eq. (5) (black line) with ${\gamma }_1\simeq 0.465$ and $\kappa \simeq 0.315$.

Figure 3

Fractal fluctuations. (a) Simulations of the scaling of $\sigma (M_c)$ with $L$ show excellent agreement with Eq. (3) where $d_f^{\prime }=3d/4$ (see inset). (b) Data collapse of $P(M_c)$ for $d=2$ under the scaling $P(M_c)L^{d_f^{\prime }}\sim \mathcal{F}[(M_c-⟨M_c⟩)/L^{d_f^{\prime }}]$ (black line), where $\mathcal{F}$ is a skewed Gaussian as in Eq. (5), now with ${\gamma }_1\simeq -0.607$ and $\kappa \simeq 0.449$. (c) Close to the hybrid percolation threshold, the MGCC’s fluctuations are fractal up to length scales (represented by $L$) below ${\xi }^{\prime }$ [Eq. (1)] and nonfractal otherwise, as described by the universal scaling function in Eq. (4). Results are shown for $d=2$ and for $d=3$ in the inset.

Figure 4

Fractal fluctuations in interdependent magnetization. (a) Mixed-order transition in randomly interdependent Ising $d$-dimensional lattices (here $d=2$, 3, 4) measured via the magnetic density $M/N$ as a function of temperature $T$. Inset: the critical temperature $T_c$ scales linearly with the lattices’ dimension $d$. (b) Simulations of the scaling of $\sigma (T_c)$ with $L$ show excellent agreement with Eq. (2) where ${\nu }^{\prime }=2/d$ as shown in the inset. (c) At criticality ${\xi }^{\prime }$ diverges and the fluctuations of the MGCC are fractal in all length scales and follow Eq. (3) with $d_f^{\prime }=3d/4$ as shown in the inset. (d) Fluctuations are fractal up to ${\xi }^{\prime }$ [Eq. (1)] and nonfractal above it, confirming the scaling in Eq. (4), here shown with $d=2$.