Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions:
$f(r) =$ number of graph nodes contained within the a radius $r$ from the centre
$g(r) =$ number of graph edges connecting a node within a radius $r$ of the centre to one outside a radius $r$ from the centre
and the limiting ratio $$c = \lim_{r \rightarrow 1} \frac{g(r)}{f(r)}$$ In the case of the link given I think the 2-in-4-out structure of the graph means that $c=2$. Is this correct? Is there a simple way of 'reading off' this limiting ratio in the case of projections of more general tilings particularly those lacking the useful symmetries of the one considered here.