Complexity_ A Guided Tour - Melanie Mitchell [137]
To run the RBN, give each node an initial state of on or off chosen at random. Then at each time step, each node transmits its state to the nodes it links to, and receives as input the states from the nodes that link to it. Each node then applies its rule to its input to determine its state at the next time step. All this is illustrated in figure 18.2, which shows the action of an RBN of five nodes, each with two inputs, over one time step.
RBNs are similar to cellular automata, but with two major differences: nodes are connected not to spatially neighboring nodes but at random, and rather than all nodes having an identical rule, each node has its own rule.
In Kauffman’s work, the RBN as a whole is an idealized model of a genetic regulatory network, in which “genes” are represented by nodes, and “gene A regulates gene B” is represented by node A linking to node B. The model is of course vastly simpler than real genetic networks. Using such idealized models in biology is now becoming common, but when Kauffman started this work in the 1960s, it was less well accepted.
LIFE AT THE EDGE OF CHAOS
Kauffman and his students and collaborators have done a raft of simulations of RBNs with different values of the in-degree K for each node. Starting from a random initial state, and iterated over a series of time steps, the nodes in the RBN change state in random ways for a while, and finally settle down to either a fixed point (all nodes’ states remain fixed) or an oscillation (the state of the whole network oscillates with some small period), or do not settle down at all, with random-looking behavior continuing over a large number of iterations. Such behavior is chaotic, in that the precise trajectory of states of the network have sensitive dependence on the initial state of the network.
Kauffman found that the typical final behavior is determined by both the number of nodes in the network and each node’s in-degree K. As K is increased from 1 (i.e., each node has exactly one input) all the way up to the total number of nodes (i.e., each node gets input from all other nodes, including itself), the typical behavior of the RBNs moves through the three different “regimes” of behavior (fixed-point, oscillating, chaotic). You might notice that this parallels the behavior of the logistic map as R is increased (cf. chapter 2). At K = 2 Kauffman found an “interesting” regime—neither fixed point, oscillating, or completely chaotic. In analogy with the term “onset of chaos” used with the logistic map, he called this regime the “edge of chaos.”
Assuming the behavior of his RBNs reflected the behavior of real genetic networks, and making an analogy with the phases of water as temperature changes, he concluded that “the genomic networks that control development from zygote to adult can exist in three major regimes: a frozen ordered regime, a gaseous chaotic regime, and a kind of liquid regime located in the region between order and chaos.”
Kauffman reasoned that, for an organism to be both alive and stable, the genetic networks his RBNs modeled had to be in the interesting “liquid” regime—not too rigid or “frozen,” and not too chaotic or “gaseous.” In his own words, “life exists at the edge of chaos.”
Kauffman used the vocabulary of dynamical systems theory—attractors, bifurcations, chaos—to describe his findings. Suppose we call a possible configuration of the nodes’ states a global state of the network. Since RBNs have a finite number of nodes, there are only a finite number of possible global states, so if the network is iterated for long enough it will repeat one of the global states it has already been in, and hence cycle through the next series of states until it repeats that global state again. Kauffman called this cycle an “attractor” of the network. By performing many