Space as a network. Is it possible to model topological properties of Euclidean (or non-Euclidean) space using a discrete network of points? I've read Wolfram's book 'A New Kind of Science', which was very interesting and introduced me to the topic of cellular automata.
But he also introduces the idea to model the physical space by a network, by which, as I understand, he means an unordered uniform (on average) connected graph with finite number of edges meeting at each vertex (the same number for each vertex, again, on average).
So, at first, his idea seems sound. Really, if we take all edges to be the length of 1, and define the distance between the points in our 'space' as the length of a shortest path between the vertices, we can recover some properties of a real space. For example, we can define a 'circle' as a set of points all at the same distance from a given point.
Then if the number of points on a 'circle' grows linearly with the distance, we get 2D space, if it grows as a square of the distance we get 3D space and so on.
Wolfram talks about some 'continuum limit' when the distances become very large, but studying some examples of uniform networks I found that there is no such limit.

The problem with uniform discrete networks I considered is the number of possible shortest paths (straight lines) between two points. In every network I considered (even with deliberately introduced randomness) the number of the 'straight lines' a. actually grows with the distance, so in the continuum limit we get an infinite number of possible straight lines between two points and b. depends on the orientation of the network.
But I considered only simple networks (square grid, hexagonal grid) with some randomness introduced.
Is it actually possible to construct a network that will be a good topological model for the Euclidean (or non-Euclidean) space?
 A: I really like the spirit of your question - what discrete spaces (with graph metrics) have euclidean space as their continuum limit?  First, I don't think you have to worry about multiple shortest length paths on the graphs.  If we think about the rectangular lattice $\mathbb{Z}^2$, the number of shortest paths between $(0,0)$ and $(1,n)$ is $n+1 \to \infty$, and this is practically the best case scenario.  It doesn't jive with the fact that in Euclidean space we have a unique shortest length path between any two points.  However, the graph metric as you've defined it in this space is 
$$ d_G((x_1,y_1),(x_2,y_2)) = |x_1 - x_2| + |y_1 - y_2|, $$
which would more naturally have (and does have) the $L^1$ norm (taxicab norm) as its limit:
$$ \| (x_1,y_1) - (x_2,y_2))\|_{L^1} = |x_1 - x_2| + |y_1 - y_2|. $$
This metric's Borel sigma-algebra is generated by the diamonds $B_{L^1}(a,r) = \{b : \|a - b\|_{L^1} < r \}$ (where $a,b$ are points in $\mathbb{R}^2$).  If we are only interested in the topology of Euclidean space, this does a fine job of recovering it; the $L^1$ basis described generates the same topology as the Euclidean topology.
Now if we hope to recover the Euclidean geometry I think we'll need to alter the graph metric, so that we end up with straight-line shortest paths between points, as opposed to the $L^1$ case.  A shortest path between $(0,0)$ and $(1,1)$ in the $L^1$ distance is given by any path moving only straight left or straight up between the two points.  To get a more familiar geometry (metric) we could make the lattice complete and weight the edges so that (embedding the graph into $\mathbb{Z}^2$) the edge lengths are given as 
$$ d_G((x_1,y_1),(x_2,y_2)) = |x_1 - x_2|^{\alpha} + |y_1 - y_2|^{\alpha}, ~~~ \alpha > 1. $$
It isn't obvious to me that we can get the classic geometry using only the graph distance.
