regular manner. But with rules (e), (f) and (g) the networks are more complicated, and begin to seem somewhat random.

The procedure that is used to lay out the networks on the previous two pages [492, 493] is a direct analog of the procedure used for space networks on page 479: the row in which a particular node will be placed is determined by the minimum number of connections that have to be followed in order to reach that node starting from the node at the top.

In cases (a) and (c) the networks obtained in this way have the property that all connections between nodes go either across or down the page. But in every other case shown, at least some connections also go up the page. So what does this mean for our notion of time? As mentioned earlier, there can never be a loop in any causal network that comes from an evolution process. But if one identifies time with position down the page, the presence of connections that go up as well as down the page implies that in some sense time does not always progress in the same direction. Yet at least in the cases shown here there is still a strong average flow down the page—agreeing with our everyday perception that time progresses only in one direction.

Like in so many other systems that we have studied in this book, the randomness that we find in causal networks will inevitably tend to wash out details of how the networks are constructed. And thus, for example, even though the underlying rules for a mobile automaton always treat space and time very differently, the causal networks that emerge nevertheless often exhibit a kind of uniform randomness in which space and time somehow work in many respects the same.

But despite this uniformity at the level of causal networks, the transformation from mobile automaton evolution to causal network is often far from uniform. And for example the pictures at the top of the facing page show the causal networks for rules (e) and (f) from the previous page—but now with each node numbered to specify the step of mobile automaton evolution from which it was derived.

And what we see is that even nodes that are close to the top of the causal network can correspond to events which occur after a large number of steps of mobile automaton evolution. Indeed, to fill in just twenty rows

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