I am currently studying appendix A.6 of the article "Bit threads and holographic entanglement".In this appendix authors present an approximation for Riemannian manifold. I am in particularly interested in applying this approach to Hyperbolic disc and so I will recap the approach directly for that case.
First let us introduce several length scales we are to use:
$$0 \ll \epsilon'' \ll \epsilon' \ll \epsilon \ll l_{\text{AdS}}$$
where $l_{\text{AdS}}$ is curvature scale of Hyperbolic disc.
First we take rectangular region of size of order $\epsilon \times \epsilon$. Since $\epsilon \ll l_{\text{AdS}}$ this region should be almost flat. Next we produce maximal packing of this region with circles of $ \epsilon''$ radius. Finally we define needles of the length $\epsilon'<\delta<1.1 \epsilon'$ and use them in order to connect all possible centers of aforementioned circles between each other.
According to the article if one takes $\epsilon'' , \frac{\epsilon''}{\epsilon'}, \epsilon',\frac{\epsilon'}{\epsilon}, \epsilon $ goes to $0$ (at this point I haven't used $\epsilon$ scale yet I've decided to include it here since it has been done so in the article) then:
This sequence of graphs appears capable of giving a reasonable representation of differential information.
Now I want to do simple check of this set up. Suppose we take two points inside aforementioned region. Length between them should be $\sim \sqrt{x^2 +y^2}$(In the sense that this is just straight line on the plane). I want to see how this may be derived from above setup.
My working idea is the following one:
Let us consider minimal cut through the graph.
Since the needles of length $\delta$ are distributed uniformly around this region we can associate certain $\overline{l_g}$ to each intersection of the needle by given cut. Then length of any minimal cut will be given by: $$L_{\text{min cut}} = \overline{l_g} \cdot N_{\text{tot}}$$
Then one could compare length of this cut with length of corresponding continuous cut. I believe that by simulation one might show that for properly defined $\overline{l_g}$ two length will almost be equivalent. The main problem that I am facing here is how to justify formulas for $L_{\text{min cut}}$ and $\overline{l_g}$ in more precise way.
If there is another way to deal with the problem I would be delighted to know about it.