Following up on the discussions on these two previous blogposts, I've put my argument why Poincaré-invariant networks in Minkowski space cannot exist into writing, or rather drawing. The notes are on the arxiv today.The brief summary goes like this: We start in 1+1 dimensions. Suppose there is a Poincaré-invariant network in Minkowski space that is not locally infinitely dense, then its nodes must be locally finite and on the average be distributed in a Poincaré-invariant way. We take this distribution of points and divide it up into space-time tiles of equal volume. Due to homogeneity, each of these volumes must contain the same number of nodes on the average. Now we pick one tile (marked in grey).
Since the same happens for each other node, and there are infinitely many nodes in the lightlike past and future of the center tile, there are infinitely many links passing through the center tile, and due to homogeneity also through every other tile. Consequently the resulting network is a) highly awkward and b) locally infinitely dense.
What does this mean? It means that whenever you are working with an approach to quantum gravity based on space-time networks or triangulations, then you have to explain how you want to recover local Lorentz-invariance. Just saying "random distribution" doesn't make the problem go away. The universe isn't Poincaré-invariant, so introducing a preferred frame is not in and by itself unreasonable or unproblematic. The problem is to get rid of it on short distances, and to make sure it doesn't conflict with existing constraints on Lorentz-invariance violations.
I want to thank all those who commented on my earlier blogposts which prompted me to write up my thoughts.