large-scale limit, with properties of ordinary space. And the notion of distance is perhaps the most fundamental of such properties.
A simple way to define the distance between two points is to say that it is the length of the shortest path between them. And in ordinary space, this is normally calculated by subtracting the numerical coordinates of the positions of the points. But on a network things become more direct, and the distance between two nodes can be taken to be simply the minimum number of connections that one has to follow in order to get from one node to the other.
But can one tell just by looking at such distances whether a particular network corresponds to ordinary space of a certain dimension?
To a large extent one can. And a test is to see whether there is a way to lay out the nodes in the network in ordinary space so that the distances between nodes computed from their positions in space agree—at least in some approximation—with the distances computed directly by following connections in the network.
The three networks at the top of the previous page were laid out precisely so as to make this the case respectively for one, two and three-dimensional space. But why for example can the second network not be laid out equally well in one-dimensional rather than two-dimensional space? One way to see this is to count the number of nodes that appear at a given distance from a particular node in the network.
And for this specific network, the answer for this is very simple: at distance r there are exactly 3r nodes—so that the total number of nodes out to distance r grows like r^2. But now if one tried to lay out all these nodes in one dimension it is inevitable that the network would have to bulge out in order to fit in all the nodes. And it turns out that it is uniquely in two dimensions that this particular network can be laid out in a regular way so that distances based on following connections in it agree with ordinary distances in space.
For the other two networks at the top of the previous page similar arguments can be given. And in fact in general the condition for a network to correspond to ordinary d-dimensional space is precisely that the total number of nodes that appear in it out to distance r grows in some limiting sense like r^d—a result analogous to the standard