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.

• The paper’s main contribution is the development of a comprehensive model that integrates feedback mechanisms into percolation theory, yielding novel dynamical behaviors such as explosive transitions.
• It employs generating function analysis and iterative maps to capture continuous, discontinuous, and chaotic transitions across diverse feedback regimes.
• The findings highlight the critical role of combined network topology and feedback functions in governing resilience and dynamic instability in complex adaptive systems.

Feedback Percolation on Complex Networks: A Formal Review

Introduction and Motivation

The study introduces a comprehensive theoretical framework for feedback percolation on complex networks, elevating percolation theory by incorporating explicit dynamical coupling between the microscopic activation probability and the macroscopic order parameter—specifically, the size of the giant component. In contrast to classical percolation, where the activation probability is static, feedback percolation endogenizes the activation process: the macroscopic state (giant component) modulates the probability of local link activation and vice versa. This formulation is motivated by real-world systems such as adaptive neural circuitry, epidemic spreading influenced by behavioral response, and self-regulating infrastructure, where global system states dynamically influence local interactions.

Figure 1: Schematic illustration of feedback percolation dynamics where the activation probability and the size of the giant component reciprocally modulate each other across iterations.

Formal Model

The dynamics are defined on a network of size $N$ where initially all links are inactive. The initial activation probability is $p_0 = p$. The process is iterative; at step $n$, the bond activation probability is recursively updated as

$p_n = p + f(S_{n-1})$

where $f$ is a feedback function of the previous step's giant component size $S_{n-1}$. At every iteration, the system updates the network configuration to ensure consistency with $p_n$ and computes $S_n$. This iterative coupling is mathematically cast as a one-dimensional delayed map $S_n = h(S_{n-1})$, endowing the system with rich nonlinear behavior. Both positive and negative feedback regimes, as well as non-monotonic cases, are examined.

The macroscopic phase structure is extracted via generating function analysis for random graphs with degree distribution $P(k)$, giving rise to self-consistency equations for $p_0 = p$0 (the probability a randomly followed link does not reach the giant component) and $p_0 = p$1. The percolation threshold is now a function of both network topology and the feedback function, generalizing the Molloy-Reed criterion.

Dynamical Phenomena and Theoretical Results

The phase behavior of feedback percolation is governed by the form of $p_0 = p$2. Three principal cases are systematically addressed: positive feedback, negative feedback, and non-monotonic feedback.

Positive Feedback

With $p_0 = p$3, positive feedback leads to reinforcement between the size of the giant component and local activation probability. The theoretical and empirical results on Erdős–Rényi (ER) graphs with mean degree $p_0 = p$4 demonstrate both continuous and discontinuous transitions. Specifically:

• There is a percolation threshold $p_0 = p$5 identical to classical percolation, but for $p_0 = p$6, feedback induces a sharp, discontinuous increase ("explosive percolation") in $p_0 = p$7 at $p_0 = p$8, a phenomenon absent in static models.
• The discontinuous transition displays hybrid criticality: $p_0 = p$9, implying a square-root singularity at the jump.
• The system can exhibit "double transitions": a continuous percolation at $n$0, immediately followed by a discontinuous transition at $n$1 (see Figure 1).

This result challenges the typical dichotomy of continuous vs. discontinuous percolation by producing rich bifurcation structures controlled by feedback strength $n$2.

Negative Feedback and Oscillatory Phases

For negative feedback $n$3, the system self-regulates, and the fixed point corresponding to a stable giant component can lose stability through a period-doubling bifurcation as $n$4 increases, leading to stable oscillations in $n$5:

• For $n$6, the negative feedback dominates, and the system enters a regime of period-2 oscillations, reflecting real-world systems where macroscopic growth triggers compensatory suppression in the next step (e.g., epidemic control, congestion protocols).
• The phase diagram confirms regions of steady, oscillatory, and non-percolating behavior as functions of $n$7 and $n$8, with boundaries analytically determined by linear stability analysis on the iterative map.

Route to Chaos under Non-monotonic Feedback

A non-monotonic, quadratic feedback function $n$9 is shown to drive the system through classic period-doubling cascades into chaos, placing the percolation dynamics within the logistic universality class:

• As $p_n = p + f(S_{n-1})$0 is increased, the bifurcation diagram displays fixed points (steady giant component), period-2 oscillations, and aperiodic (chaotic) evolution of the system’s largest component.
• Numerical calculations of the Lyapunov exponent confirm positive values in the chaotic regime, substantiating sensitive dependence on initial conditions.

The results thus position feedback percolation as a minimal model for dynamical and even unpredictable macroscopic states in networked systems with delayed global regulatory mechanisms.

Relation to Classical and Multilayer Percolation

Size-inverted negative feedback ($p_n = p + f(S_{n-1})$1), where suppression is strongest when the giant component is small, recapitulates the physics of cascading failures in interdependent multilayer networks. Specifically, the self-consistency equations in this feedback regime map directly onto those for two-layer ER interdependent networks, including the emergent hybrid phase transition, reinforcing the model’s generality and explanatory power for systemic collapse dynamics.

Implications and Future Directions

The introduction of global feedback into percolation fundamentally alters the qualitative and quantitative nature of percolative transitions on networks. The work highlights that:

• The detailed shape and strength of the feedback function critically determine whether the system exhibits classical criticality, explosion, self-regulation, or chaotic fluctuations.
• Feedback percolation offers a unifying framework for a wide array of phenomena, from adaptive biological and socio-technical networks to the catastrophic failure of infrastructures with tightly coupled components.
• These findings imply that resilience and controllability in complex systems are deeply sensitive not only to topology but also to macroscopic-microscopic regulatory couplings.

Future exploration can systematically extend the framework to multiplex, higher-order, and temporal networks, potentially connecting to adaptive network theory, synchronization, and co-evolutionary dynamics. The logistic universality in the route to chaos suggests connections with the broader theory of nonlinear maps and critical phenomena.

Conclusion

Feedback percolation provides a rigorous, generalizable extension of percolation theory that accounts for the dynamical co-dependence between local connectivity and global order. The framework yields a comprehensive picture of emergent phenomena—explosive growth, hybrid criticality, limit cycles, and chaos—not accessible to static models. These results have significant theoretical and practical implications, particularly for understanding the resilience, adaptability, and dynamical instability of real-world complex systems.