Many mathematical results exist about continuous topolog-ical surfaces of negative curvature. We give here some properties of dis-crete regular tessellations on such objects and explain a characterization of discrete geodesics and areas that shows how such hyperbolic networks can be seen as intermediary structures between Euclidean infinite tessel-lations (like square grid) and regular infinite trees. We do not use some possible group structures of this networks (Cayley graphs) but only geometrical arguments in our constructive proofs. Hence we can see that there are few geodesics in hyperbolic networks and that large areas have very unsmooth borders.