successively greater network distances appear in successive columns across the page. And this setup has the feature that the height of column r gives the number of nodes that are at network distance r.
So by looking at how these heights grow across the page, one can see whether there is a correspondence with the r^(d-1) form that one expects for ordinary d-dimensional space. And indeed in case (g), for example, one sees exactly r1 linear growth, reflecting dimension 2.
Similarly, in case (d) one sees r0 growth, reflecting dimension 1, while in case (h) one sees r2 growth, reflecting dimension 3.
Case (f) illustrates slightly more complicated behavior. The basic network in this case locally has an essentially two-dimensional form—but at large scales it is curved by being wrapped around a sphere. And what therefore happens is that for fairly small r one sees r1 growth—reflecting the local two-dimensional form—but then for larger r there is slower growth, reflecting the presence of curvature.
Later in this chapter we will see how such curvature is related to the phenomenon of gravity. But for now the point is just that network (f) again behaves very much like ordinary space with a definite dimension.
So do all sufficiently large networks somehow correspond to ordinary space in a certain number of dimensions? The answer is definitely no. And as an example, network (i) from the previous page has a tree-like structure with 3r nodes at distance r. But this number grows faster than rd for any d—implying that the network has no correspondence to ordinary space in any finite number of dimensions.
If the connections in a network are chosen at random—as in case (j)—then again there will almost never be the kind of locality that is needed to get something that corresponds to ordinary finite-dimensional space.
So what might an actual network for space in our universe be like?
It will certainly not be as simple and regular as most of the networks on the previous page. For within its pattern of connections must be encoded everything we see in our universe.
And so at the level of individual connections, the network will most likely at first look quite random. But on a larger scale, it must be arranged so as to correspond to ordinary three-dimensional space. And somehow whatever rules update the network must preserve this feature.