Feedback percolation on complex networks

Abstract

Traditional percolation theory assumes static microscopic rules, limiting its ability to describe real-world complex systems where macroscopic order actively regulates local interactions. Here, we introduce feedback percolation, an unified framework that dynamically couples the microscopic activation probability to the macroscopic size of the giant component. We show that this simple feedback mechanism produces a rich variety of behaviors both analytically and numerically. Depending on the feedback functions, the system exhibits explosive discontinuous jumps, hybrid transitions, limit-cycle oscillations, and routes to chaos, absent in classical percolation. Our findings establish that macroscopic feedback provides a unifying physical mechanism for phenomena ranging from self-regulating oscillations to systemic infrastructure collapse.

Figures (7)

• Figure 1: (a) Schematic illustration of feedback in percolation models. The activation probability and the size of the giant component influence each other. (b) An iterative process of the feedback mechanism where the activation probability is updated as $p_{n}=p+f(S_{n-1})$. Here, $p_n$ and $S_n$ represent the activation probability and the size of the giant component after $n$ iterations of the feedback process, respectively. Solid and dotted lines indicate active and inactive links, respectively, and the largest component is highlighted in gray. The newly activated link due to the feedback is marked in red.
• Figure 2: (a) The size of the giant component, $S_\infty$, under positive feedback as a function of $p$ on ER graphs with $z=4$. Symbols represent numerical results for $N=10^5$, and solid lines represent theoretical predictions. (b) The number $T$ of iterations required to reach the steady state in numerical simulations as a function of $p$. The inset displays the average size of finite connected components. (c) Phase diagram in the $(p,q)$ plane for the positive feedback model. The solid line indicates the location of continuous percolation transitions, while the dashed line indicates that of a discontinuous jump of $S_\infty$. The inset shows the interaction between $\phi_p(S)$ and $S$ for $q=0.2$.
• Figure 3: (a) Bifurcation diagram under negative feedback with $q=0.2$ on ER graphs with $z=4$. Numerical results (symbols) for $N=10^5$ and theoretical predictions (lines) are shown together. (b) Phase diagram in the $(p,q)$ plane for the negative feedback model. The solid green line represents the boundary $p_c$ of the percolation transitions, and the red dashed line indicates the boundary $p_o$ of the onset of oscillatory behavior. The inset shows the time course of the size $S_n$ of the giant component.
• Figure 4: (a) Bifurcation diagram under the non-monotonic feedback on ER graphs with $z=4$. Numerical results (red) for $N=10^5$ and the theoretical results (gray) are shown together. The inset shows the time course of the size $S_n$ of the giant component. (b) Cobweb plot within a chaotic regime: $(p,q)=(0.54,1)$. The red line represents the iterative map, and the green line represents $S_n=S_{n-1}$.
• Figure 5: (a) The size of the giant component $S_\infty$ under the size-inverted negative feedback as a function of $p$ on ER graphs with $z=4$. Theoretical results (lines) show agreement with numerical results (symbols) for $N=10^5$. (b) Phase diagram in the $(p,q)$ plane for the size-inverted negative feedback model. (inset) The square of the giant component size under the feedback with $q=1$, i.e., $p_n=pS_{n-1}$, on ER graphs with $z=4$ and $6$. Numerical results (symbols) based on the feedback model with $N=10^5$ show agreement with theoretical results (lines) for the cascading dynamics on interdependent networks.