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But in fact, such a network has vastly less information. For given just a set of connections between nodes, there is no obvious way even to know which of these connections should be associated with the same face—let alone to work out anything like angles between faces.

Yet despite this, it turns out that all the geometrical features that are ultimately of relevance to general relativity can actually be determined in large networks just from the connectivity of nodes.

One of these is the value of the so-called Ricci tensor, which in effect specifies how the Ricci scalar curvature is made up from different curvature components associated with different directions.

As indicated above, the scalar curvature associated with a network is directly related to how many nodes lie within successive distances r of a given node on the network—or in effect how many nodes lie within successive generalized spheres around that node. And it turns out that the projection of the Ricci tensor along a particular direction is then just related to the number of nodes that lie within a cylinder oriented in that direction. But even just defining a consistent direction in a network is not entirely straightforward. But one way to do it is simply to pick two points in the network, then to say that paths in the network are going in the same direction if they are segments of the same shortest path between those points. And with this definition, a region that approximates a cylinder can be formed just by setting up spheres with centers at every point on the path.

But there is now another issue to address: at least in its standard formulation general relativity is set up in terms of properties not of three-dimensional space but rather of four-dimensional spacetime. And this means that what is relevant are properties not so much of specific networks representing space, but rather of complete causal networks.

And one immediate feature of causal networks that differs from space networks is that their connections go only one way. But it turns out that this is exactly what one needs in order to set up the analog of a spacetime Ricci tensor. The idea is to start at a particular event in the causal network, then to form what is in effect a cone of events that can be reached from there. To define the spacetime Ricci tensor, one considers—as on page 516—a sequence of spacelike slices through this

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