But how can we determine the curvature from the structure of each network? Earlier in this chapter we saw that if a network is going to correspond to ordinary space in some number of dimensions d, then this means that by going r connections from any given node one must reach about rd-1 nodes. But it turns out that when curvature is present it leads to a systematic correction to this.
In each of the pictures on the facing page the network shown can be thought of as corresponding to two-dimensional space. And this means that to a first approximation the number of nodes reached must increase linearly with r. But the bottom row of pictures show that there are corrections to this. And what happens is that when there is positive curvature—as in the pictures on the left—progressively fewer than r nodes end up being reached. But when there is negative curvature—as on the right—progressively more nodes end up being reached. And in general the leading correction to the number of nodes reached turns out to be proportional to the curvature multiplied by rd+1.
So what happens in more than two dimensions? In general the result could be very complicated, and could for example involve all sorts of different forms of curvature and other characteristics of space. But in fact the leading correction to the number of nodes reached is always quite simple: it is just proportional to what is called the Ricci scalar curvature, multiplied by rd+1. And already here this is some suggestion of general relativity—for the Ricci scalar curvature also turns out to be a central quantity in the Einstein equations.
But in trying to see a more detailed correspondence there are immediately a variety of complications. Perhaps the most obvious is that the traditional mathematical formulation of general relativity seems to rely on many detailed properties of continuous space. And while one expects that sufficiently large networks should in some sense act on average like continuous space, it is far from clear at first how the kinds of properties of relevance to general relativity will emerge.
If one starts, say, from an ordinary continuous surface, then it is straightforward to approximate it as in the picture below by a collection of flat faces. And one might think that the edges of these faces would define a network of the kind I have been discussing.
A surface approximated by flat faces whose edges form a trivalent network.