When does a system absorb a disturbance, when does it get lost in noise, and when does it settle into a new order?
In 1969 Stuart Kauffman asked what happens in genetic networks built entirely at random. His answer, later developed at the Santa Fe Institute, is that order can appear for free, and that the most interesting behavior lives on the boundary between frozen order and chaos. Here you can build these networks, poke them, and watch what they do with the poke.
N nodes, each on or off. Each node reads K nodes of the network and computes its next state with a randomly drawn rule; p is the probability that a rule outputs “on”. All nodes update at once.
Because the number of possible states is finite, the network eventually repeats itself and falls into an attractor: a fixed point or a cycle. Kauffman proposed reading attractors as cell types: one genome, several stable fates.
Perturb = flip one node by hand, against what the network dictates.
Then watch whether the perturbation is absorbed (the network returns to the same attractor), moves the system to a different attractor that then holds on its own, or spreads until the history is lost.
This is a toy model. Its exact numbers did not survive later research, but its central question did.
λ = 2·p·(1−p)·K — average sensitivity: how many nodes, on average, one changed node changes at the next step.
Let the network run until it finds its attractor. Then press Perturb, or tap a node on the circle, and watch: does the system come home or move house? To the right of the circle, each row is a node and each column a moment; nodes that stopped changing are drawn faint. They form the frozen core.
The same network runs twice. The two worlds start identical except for a few nodes. In amber, the places where they differ: the footprint of the perturbation, which physicists call damage. Below, the distance between the worlds (the fraction of the network that differs) over time, next to the prediction of Derrida and Pomeau (1986).
Now many networks at once. For each K, dozens of random networks are built, one node is flipped, and we measure what is left of the perturbation after 60 steps. Bars show the share of perturbations that died out; dots show how much of the network ended up changed. The dashed line is the theoretical prediction of Derrida and Pomeau (1986).