A Network That Decides When to Stop
A crowd grows: each new person joins the group with a probability that depends on how large the group already is. If the group is small, joining is easy; if it is huge, people start to hesitate, and growth slows down. This mechanism, called feedback, changes the entire behavior of the network.
The researchers added this loop to a mathematical model and saw a surprising diversity: the network can explode in a jump, transition smoothly to a new state, then oscillate like a pendulum, or even descend into chaos. Such regimes resemble the behavior of real systems — from power grid overloads to the spread of epidemics.
The work shows that macroscopic order can govern microscopic rules, and this is a general principle for self-regulating systems. Now we can better predict when infrastructure will hold steady and when it will collapse in an avalanche.
This approach brings together disparate phenomena under one roof and provides a tool for analyzing complex networks where the whole influences the parts.
as written by the authors arXiv:2603.22089
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.
In the feedback model, a network can behave like a predator and prey: the size of the giant component grows, then falls, and the cycle repeats endlessly, even though no external forces intervene.
In Stanislaw Lem's novel 'Solaris', the planet-ocean self-regulates through feedback, generating unexpected structures — similar to networks that change their own growth rules.