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Inevitably there is a certain arbitrariness in the way these pictures are drawn. For the underlying rules specify only what the pattern of connections in a network should be—not how its nodes should be laid out on the page. And in the effort to make clear the relationship between networks obtained on different steps, even identical networks can potentially be drawn somewhat differently.

With rule (a), however, it is fairly easy to see that a simple nested structure is produced, directly analogous to the one shown on page 509. And with rule (b), obvious repetitive behavior is obtained.

So what about more complicated behavior? It turns out that even with rule (c), which is essentially just a combination of rules (a) and (b), significantly more complicated behavior can already occur.

The picture below shows a few more steps in the evolution of this rule. And the behavior obtained never seems to repeat, nor do the networks produced exhibit any kind of obvious nested form.

What about other schemes for applying replacements? The pictures on the facing page show what happens if at each step one allows not just a single replacement, but all replacements that do not overlap.

It takes fewer steps for networks to be built up, but the results are qualitatively similar to those on the previous page: rule (a) yields a nested structure, rule (b) gives repetitive behavior, while rule (c) produces behavior that seems complicated and in some respects random.

Captions on this page:

More steps in the evolution of rule (c) from the previous page. The number of nodes increases irregularly (though roughly linearly) with successive steps.

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