Feedback Percolation in Complex Systems

• Feedback percolation is a dynamic process where microscopic occupation rules evolve based on macroscopic states, yielding complex connectivity transitions.
• It applies to diverse fields like economics, social media, reactor physics, and topological systems, explaining phenomena such as abrupt transitions, oscillations, and chaos.
• The framework reveals that closing feedback loops between micro-level dynamics and macro-scale order parameters can induce hybrid bifurcations and stabilize or destabilize networks.

Feedback percolation denotes a class of percolative processes in which microscopic occupation, activation, failure, or transport rules are not fixed ex ante, but are regulated by macroscopic state variables or by coevolving mesoscopic observables. In contrast to classical percolation, where a control parameter such as bond or site occupation probability is externally prescribed and static, feedback percolation introduces state dependence: the giant component can modify future link activation, distress can raise interest rates and thereby enlarge the susceptible population, user activity can increase follower acquisition and hence future participation probabilities, and conductance fields can evolve jointly with connectivity. Across economics, adaptive complex networks, social media, non-Hermitian topological systems, reactor physics, porous media, and interdependent networks, this modification produces phenomena that include multiple fixed points, hybrid and discontinuous transitions, self-healing regimes, limit cycles, and chaos (Jang et al., 23 Mar 2026, Solomon et al., 2014, Xie et al., 2021).

1. Conceptual scope and defining ingredients

The common structural feature is a closed causal loop between a microscopic percolation variable and a macroscopic order parameter. In the most explicit network formulation, the microscopic edge-activation probability becomes a function of the giant-component size, $p_n=p+f(S_{n-1})$, so that percolation and regulation co-evolve at each iteration (Jang et al., 23 Mar 2026). In the economic formulation of Minsky instability, the top-down variable is the interest rate $i$, the bottom-up variables are either loan demand or the number of ponzi firms, and peer-to-peer contagion propagates across firm-firm network links (Solomon et al., 2014). In the social-media formulation, participation probabilities are heterogeneous because node occupation is weighted by activity, and activity itself grows with follower count (Xie et al., 2021).

Domain	Microscopic rule under feedback	Macroscopic regulator
Complex networks	$p_n = p + f(S_{n-1})$	Giant component size $S_n$
Financial contagion	Failures and ponzi status depend on $i_t$	Interest rate $i_{t+1}=i_0 N_t^\alpha$
Social media	Node selection weighted by $q_i=m_i/\sum_j m_j$	Follower-activity coevolution

A second recurring ingredient is threshold behavior inherited from percolation theory but reshaped by feedback. In the financial model, a critical fragility density $\rho_C$ governs when failures can span the network (Solomon et al., 2014). In adaptive network models, bifurcation conditions determine discontinuous jumps or flip instabilities (Jang et al., 23 Mar 2026). In social media, positive feedback lowers the effective percolation threshold relative to uniform site percolation (Xie et al., 2021). This suggests that feedback percolation is best understood not as a single canonical equation, but as a family of state-dependent percolation mechanisms.

2. Order-parameter-controlled percolation on complex networks

A general dynamical framework is given by feedback percolation on random graphs, where the usual occupation parameter becomes endogenous (Jang et al., 23 Mar 2026). For a degree distribution $P(k)$ with generating functions

$$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$

the iterative dynamics is

$i$0

The resulting system is a one-dimensional delayed map $i$1, and fixed points satisfy

$i$2

In the absence of feedback, the ordinary threshold is recovered from linearization at $i$3:

$i$4

The framework distinguishes several feedback laws. For positive power-law feedback,

$i$5

the system on an Erdős–Rényi graph with mean degree $i$6 exhibits a continuous transition at $i$7 and, for weak feedback, a second hybrid discontinuous jump at $i$8 with square-root scaling $i$9 near the jump. As $p_n = p + f(S_{n-1})$0 increases, the continuous and discontinuous transitions approach and coalesce at the critical endpoint $p_n = p + f(S_{n-1})$1; above that point the only transition is discontinuous (Jang et al., 23 Mar 2026).

For negative power-law feedback,

$p_n = p + f(S_{n-1})$2

the continuous threshold at $p_n = p + f(S_{n-1})$3 persists, but the fixed point can lose stability at $p_n = p + f(S_{n-1})$4 through a flip bifurcation, producing a stable period-2 oscillation of $p_n = p + f(S_{n-1})$5. For sufficiently strong negative feedback, oscillation appears immediately for $p_n = p + f(S_{n-1})$6 (Jang et al., 23 Mar 2026). For non-monotonic feedback,

$p_n = p + f(S_{n-1})$7

the model undergoes a flip bifurcation and then a period-doubling cascade into chaos; the map is unimodal with a quadratic maximum and lies in the logistic universality class, with numerically positive Lyapunov exponent $p_n = p + f(S_{n-1})$8 in the chaotic regime (Jang et al., 23 Mar 2026).

The analytical bifurcation conditions make the departure from classical percolation explicit. A saddle-node discontinuity occurs when

$p_n = p + f(S_{n-1})$9

whereas period-2 onset occurs when

$S_n$0

These criteria isolate the mechanism by which macroscopic feedback creates phenomena absent in static percolation.

3. Financial instability, interscale coupling, and contagion

In the Minsky framework, feedback percolation appears as the interaction of top-down, bottom-up, and peer-to-peer positive feedback loops (Solomon et al., 2014). The top-down variable is the interest rate $S_n$1, which plays the role of a “price” set by banks in response to observed distress. Before the Minsky moment, the relevant bottom variable is the total quantity of new loans,

$S_n$2

After the Minsky moment, the relevant bottom variable is the number of ponzi firms,

$S_n$3

Firm-level heterogeneity is represented by resilience

$S_n$4

and the ponzi fraction at interest rate $S_n$5 is

$S_n$6

Peer-to-peer contagion is introduced through a network of trade or credit links. A ponzi firm fails openly only if at least one neighbor has already failed. Near the critical ponzi density $S_n$7, the average contagion-cluster size obeys

$S_n$8

with $S_n$9 and seed-dependent constant $i_t$0. Below $i_t$1, failure clusters remain finite; as $i_t$2, they diverge into a giant cluster (Solomon et al., 2014).

The top-down reaction of banks is modeled by

$i_t$3

and the network-corrected failure count is

$i_t$4

Without network effects, the coupled equations for $i_t$5 and $i_t$6 yield a single fixed point

$i_t$7

with log-space eigenvalues $i_t$8. The fixed point is convergent for $i_t$9 and repelling for $i_{t+1}=i_0 N_t^\alpha$0 (Solomon et al., 2014).

With network effects, up to three fixed points typically appear: $i_{t+1}=i_0 N_t^\alpha$1 as a stable attractor, $i_{t+1}=i_0 N_t^\alpha$2 as an unstable repeller, and $i_{t+1}=i_0 N_t^\alpha$3 as a “very-solid-core” attractor in which only the most resilient firms survive. The phase diagram contains four regimes: micro-crisis, stable self-healing, Minsky instability, and solid-core. The paper’s narrative of feedback percolation is an autocatalytic sequence: a small shock raises failures, banks raise $i_{t+1}=i_0 N_t^\alpha$4, more firms become ponzi, nearby fragile firms fail by percolation, and the loop iterates until the system either self-limits or enters systemic crisis (Solomon et al., 2014).

4. Coevolutionary information cascades on social media

In large social-media networks, feedback percolation was introduced to explain why real information cascades occur at far lower retweet probabilities than uniform percolation predicts (Xie et al., 2021). The central empirical mechanism is a positive-feedback loop between follower count and activity. Over a time window $i_{t+1}=i_0 N_t^\alpha$5, the average tweeting or retweeting rate of user $i_{t+1}=i_0 N_t^\alpha$6 is

$i_{t+1}=i_0 N_t^\alpha$7

with measured values $i_{t+1}=i_0 N_t^\alpha$8 in Weibo 2012, rising to $i_{t+1}=i_0 N_t^\alpha$9 by 2014, and $q_i=m_i/\sum_j m_j$0 in Twitter. Follower growth satisfies

$q_i=m_i/\sum_j m_j$1

with $q_i=m_i/\sum_j m_j$2 in Weibo and $q_i=m_i/\sum_j m_j$3 in Twitter. More followers therefore imply higher activity, and higher activity implies faster follower gain (Xie et al., 2021).

The percolation model is heterogeneous site percolation rather than uniform occupation. Exactly $q_i=m_i/\sum_j m_j$4 nodes are occupied, but node $q_i=m_i/\sum_j m_j$5 is selected with probability

$q_i=m_i/\sum_j m_j$6

High-activity, high-degree users are therefore much more likely to participate. Using the generating function of the remaining nodes,

$q_i=m_i/\sum_j m_j$7

the expected cascade size is

$q_i=m_i/\sum_j m_j$8

with self-consistency

$q_i=m_i/\sum_j m_j$9

Uniform percolation gives $\rho_C$0, whereas feedback percolation yields a modified threshold

$\rho_C$1

so that even modest positive feedback lowers the threshold substantially (Xie et al., 2021).

Empirically, the observed critical retweet probability is approximately $\rho_C$2 in Weibo and approximately $\rho_C$3 in Twitter, while the uniform threshold predicted by $\rho_C$4 is approximately $\rho_C$5. Uniform percolation therefore grossly underestimates global outbreaks: $\rho_C$6 of true outbreaks lie below $\rho_C$7. The data-driven feedback model reproduces both the location of $\rho_C$8 and the functional form $\rho_C$9. The same feedback also intensifies influence imbalance: in Weibo 2014, the top $P(k)$0 of users generate $P(k)$1 of all outbreak-scale cascades, whereas uniform theory would place that breakpoint at approximately $P(k)$2 (Xie et al., 2021).

5. Physical realizations in topological, reactor, and transport systems

In a non-Hermitian bilayer Chern model, feedback percolation arises from the interplay of chiral topology, directed gain or loss, and interlayer tunneling (Yang et al., 2023). A random height field $P(k)$3 defines “Chern” islands via thresholding, producing site percolation with occupation probability

$P(k)$4

In the uncorrelated limit, the square-lattice threshold is $P(k)$5; for finite correlation length $P(k)$6, numerics give $P(k)$7–$P(k)$8. PT symmetry breaks only when an island width exceeds

$P(k)$9

and the edge amplitude then obeys

$$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$0

The PT-broken phase is therefore activated precisely when percolation produces sufficiently wide islands, i.e. when $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$1 (Yang et al., 2023).

In reactor physics, neutron branching random walks with Doppler feedback map onto directed percolation (Dechenaux et al., 2022). The local law is

$$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$2

where the quadratic term encodes Doppler feedback. The absorbing state is $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$3, the active state is $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$4, the order parameter is $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$5, and the control parameter is $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$6. The corresponding field theory is Reggeon field theory. At criticality, the dispersion relation becomes super-diffusive,

$$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$7

with $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$8 for $$G_0(x)=\sum_k P(k)x^k,\qquad G_1(x)=\frac{G_0'(x)}{G_0'(1)},$$9. In $i$00, the reported exponents are $i$01, $i$02, and $i$03, departing from mean-field values $i$04, $i$05, and $i$06 (Dechenaux et al., 2022).

In evolving disordered media, the term is used for conductance networks whose local bond conductances evolve with the control parameter $i$07 rather than remaining fixed (Berg et al., 2023). On the square lattice, both evolving models retain the connectivity threshold $i$08, but the conductivity exponent changes. Model I,

$i$09

remains in the classical conductivity universality class with $i$10. Model II,

$i$11

yields non-universal exponents: $i$12–$i$13 and $i$14. Here feedback is the two-way coupling between the evolving conductance distribution and macroscopic flow or connectivity (Berg et al., 2023).

6. Feedback-dependency in interdependent and interacting networks

A closely related literature studies coupled networks under a “feedback condition,” meaning that dependency links are allowed to form loops (Gao et al., 2013, Dong et al., 2013). In a network of networks, if a node in network $i$15 depends on a node in network $i$16, and dependency cycles are permitted, damage can propagate outward and then return to the original network. The self-consistency equation under feedback coupling is

$i$17

For identical networks with degree of interdependence $i$18 and coupling $i$19, this reduces to

$i$20

For Erdős–Rényi networks,

$i$21

and above

$i$22

even $i$23 cannot sustain a giant component. In the ER feedback case, the system exhibits only a continuous transition and collapse; the first-order regime present in the no-feedback condition is absent (Gao et al., 2013).

For interacting networks with both connectivity and feedback-dependency links, the mutually connected giant components $i$24 and $i$25 satisfy

$i$26

$i$27

In the symmetric case,

$i$28

and the continuous threshold is

$i$29

As coupling increases, the model passes from second-order to hybrid to first-order transitions, with the feedback case systematically less robust than the non-feedback case (Dong et al., 2013).

This literature also connects directly to later generalized feedback-percolation theory. The “size-inverted negative” law

$i$30

maps exactly onto mutual percolation on two interdependent Erdős–Rényi networks through the transformation $i$31, where

$i$32

That mapping makes explicit that classical interdependence cascades can be written as a particular feedback-percolation map (Jang et al., 23 Mar 2026).

7. Shared mechanisms, interpretive issues, and common misconceptions

The literature suggests that “feedback percolation” is not a single universally standardized formalism, but a set of related constructions unified by state-dependent microscopic rules. In some papers the feedback variable is directly the order parameter $i$33 (Jang et al., 23 Mar 2026); in others it is an endogenous control parameter such as the interest rate $i$34 (Solomon et al., 2014), a coevolving activity field $i$35 (Xie et al., 2021), a dependency loop across networks (Gao et al., 2013), or an evolving conductance law $i$36 (Berg et al., 2023). The commonality lies in closing a loop between macroscopic organization and microscopic percolation dynamics.

A common simplification is to equate feedback percolation with abrupt collapse. The cited models do not support that identification. Positive feedback can indeed generate explosive or hybrid jumps (Jang et al., 23 Mar 2026), and feedback-dependency can make interdependent networks extremely vulnerable (Gao et al., 2013, Dong et al., 2013). But negative feedback can generate stable period-2 oscillations rather than monotone cascades (Jang et al., 23 Mar 2026), Minsky networks can remain in micro-crisis or stable self-healing phases (Solomon et al., 2014), and evolving conductance networks can preserve the classical universality class despite local feedback in conductance evolution (Berg et al., 2023).

Another misconception is that feedback always changes the universality class. Again, the results are mixed. Model I of evolving disordered media retains the standard conductivity exponent $i$37 (Berg et al., 2023), whereas reactor dynamics at directed-percolation criticality departs from mean field and yields a fractional-Laplacian description with $i$38 in $i$39 (Dechenaux et al., 2022). In social media, feedback primarily lowers the threshold and intensifies heterogeneity (Xie et al., 2021); in adaptive random networks, it can instead create new bifurcation classes such as flip instabilities and chaos (Jang et al., 23 Mar 2026).

Taken together, these results place feedback percolation at the intersection of percolation theory, nonlinear dynamics, and adaptive network science. Its distinctive contribution is to replace static occupation rules by endogenous regulation, thereby turning percolation from a one-parameter connectivity problem into a dynamical theory of coevolution, instability, and self-organization across scales.