The largest damping force is the air resistance on the mat. "Fly mats" are mats made from 4 mm wide webbing on 10 mm spacing. Or made from pairs of string about 1.5 mm wide per pair on 4 mm spacing. The net effect is that it's a net. With over 50% holes.
One thing you notice on first jumping on a fly mat: You don't lose energy as fast. If you are used to losing a third of your energy doing a doggie drop, and then do this same trick on a fly mat and get 90% of your energy back it will scare the pants off of you.
Good mats are not stretchy. You want all the energy storage to be in the springs, which are better at returning it. Most elastics fabrics are very poor at returning energy -- 40-60% especially in directions not along the weave.