Network Systems

One feature of systems like cellular automata is that their elements are always set up in a regular array that remains the same from one step to the next. In substitution systems with geometrical replacement rules there is slightly more freedom, but still the elements are ultimately constrained to lie in a two-dimensional plane.

Indeed, in all the systems that we have discussed so far there is in effect always a fixed underlying geometrical structure which remains unchanged throughout the evolution of the system.

It turns out, however, that it is possible to construct systems in which there is no such invariance in basic structure, and in this section I discuss as an example one version of what I will call network systems.

A network system is fundamentally just a collection of nodes with various connections between these nodes, and rules that specify how these connections should change from one step to the next.

At any particular step in its evolution, a network system can be thought of a little like an electric circuit, with the nodes of the network corresponding to the components in the circuit, and the connections to the wires joining these components together.

And as in an electric circuit, the properties of the system depend only on the way in which the nodes are connected together, and not on any specific layout for the nodes that may happen to be used.

Of course, to make a picture of a network system, one has to choose particular positions for each of its nodes. But the crucial point is that these positions have no fundamental significance: they are introduced solely for the purpose of visual representation.

In constructing network systems one could in general allow each node to have any number of connections coming from it. But at least for the purposes of this section nothing fundamental turns out to be lost if one restricts oneself to the case in which every node has exactly two outgoing connections—each of which can then either go to another node, or can loop back to the original node itself.

With this setup the very simplest possible network consists of just one node, with both connections from the node looping back, as