Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

[State networks for] systems of limited size

For any system with a limited total number of states, it is possible to create a finite network that gives a global representation of the behavior of the system. The idea of this network (which is very different from the finite automata networks discussed above) is to have each node represent a complete state of the system. At each step in the evolution of the system, every state evolves to some new state, and this process is represented in the network by an arc that joins each node to a new node. The picture below gives the networks obtained for systems of the kind shown on page 255. Each node is labelled by a possible position for the dot. In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next.

The pictures below give networks obtained from the system shown on page 257 for various values of n. For odd n, the networks consist purely of cycles. But for even n, there are also trees of states that lead to these cycles.

In general, any network that represents the evolution of a system with definite rules will have the same basic form. There are cycles which contain states that are visited repeatedly, and there can also be trees that represent transient states that can each only ever occur at most once in the evolution of the system.

The picture below shows the network obtained from a class 1 cellular automaton (rule 254) with 4 cells and thus 16 possible states. All but one of these 16 states evolve after at most two steps to state 15, which corresponds to all cells being black.

The pictures below show networks obtained when more cells are included in the cellular automaton above. The same convergence to a single fixed point is observed.

The pictures below give corresponding results for a class 2 cellular automaton (rule 132). The number of distinct cycles now increases with the size of the system. (As discussed below, identical pieces of the network are often related by symmetries of the underlying cellular automaton system.)

In class 3, larger cycles are usually obtained, and often the whole network is dominated by a single largest cycle. The second set of pictures below summarize the results for some larger cellular automata. Each distinct region corresponds to a disjoint part of the network, with the area of the region being proportional to the number of nodes involved. The dark blobs represent cycles. (See page 1087.)

For large sizes there is a rough correspondence with the infinite size case, but many features are still different. (To recover correct infinite size results one must increase size while keeping the number of steps of evolution fixed; the networks shown above, however, effectively depend on arbitrarily many steps of evolution.)

From Stephen Wolfram: A New Kind of Science [citation]