Microtransition Cascades to Percolation

We report the discovery of a discrete hierarchy of microtransitions occurring in models of continuous and discontinuous percolation. The precursory microtransitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.

Figure 1

Discontinuous BFW percolation. A typical realization of the relative size of the largest component $C_1/N$ as a function of link density $p$ for the BFW model with $\alpha =0.1$, 0.3, 0.6, and for the continuous ER model. Inset: Discontinuous global competition model. System size $N={10}^6$.

Figure 2

Microtransition cascade to percolation in the BFW model. (a) Relative variance $R_v$ versus link density $p$ showing sharp microtransitions before $p_c$. Peaks after $p_c$ result from unstable giant components, discussed elsewhere [33]. (b) The typical evolution (and collapse) of $C_1$ versus $p$, showing jumps when $C_1\to C_1+1$. Inset of (b): ${\mathcal{R}}_v$ versus $p$ for continuous ER percolation, shown for three different system sizes. This reveals a spectrum of microresonances before $p_c=1/2$, that, in contrast to the BFW model, disappears as $N\to \infty$. All data shown are the average over 1000 realizations.

Figure 3

Scaling laws and convergence to $p_c$ for the BFW model. (a) The positions $p_i$ of the microtransitions are well fitted by Eq. (3) for $\alpha =0.1$, 0.3, 0.6. Note that we display a log-log plot suggesting $\log (p_{i+1}/p_i)=A{i}^{-b}$. Inset of (a): Evidence for $p_i\to p_c$ for $\alpha =0.1$, 0.3, 0.6 and $N={10}^7$. (b) Exponent $\sigma$ defined in Eq. (8) for $\alpha =0.1$, 0.3, 0.6 and $N={10}^7$.

Figure 4

Scaling relations for continuous percolation. Numerical evidence for the prediction $[(p_c-p_i)/(p_c-p_{i-1})]\sim [i/(i+1{)]}^{\sigma }$ from Eq. (8), for the ER model ($\sigma =1/2$, $N=2^{25}$, 30 000 realizations) and 2D lattice ($\sigma =36/91$, $N=1024\times 1024$, 30 000 realizations).