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normally be considered nearby in three-dimensional space. And so when clusters of nodes that are nearby with respect to connections on the network get updated, they can potentially propagate effects to what might be considered distant points in space.

Nevertheless, if a network is going to correspond to space as it seems to exist in our universe, such phenomena must not be too important—and in the end there must to a good approximation be the kind of straightforward locality that exists for example in the simple causal network of page 518.

In the next section I will discuss how actual physical entities like particles propagate in systems represented by causal networks. But ultimately the whole point of causal networks is that their connections represent all possible ways that effects propagate. Yet these connections are also what end up defining our notions of space and time in a system. And particularly in a causal network as regular as the one on page 518 one can then immediately view each connection in the causal network as corresponding to an effect propagating a certain distance in space during a certain interval in time.

So what about a more complicated causal network? One might imagine that its connections could perhaps represent varying distances in space and varying intervals in time. But there is no independent way to work out distance in space or interval in time beyond looking at the connections in the causal network. So the only thing that ultimately makes sense is to measure space and time taking each connection in the causal network to correspond to an identical elementary distance in space and elementary interval in time.

One may guess that this elementary distance is around 10^-35 meters, and that the elementary time interval is around 10^-43 seconds. But whatever these values are, a crucial point is that their ratio must be a fixed speed, and we can identify this with the speed of light. So this means that in a sense every connection in a causal network can be viewed as representing the propagation of an effect at the speed of light.

And with this realization we are now close to being able to see how the kinds of systems I have discussed must almost inevitably succeed in reproducing the fundamental features of relativity theory.

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