The weight of wood doesn’t correlate with the rigidity.
Wood's stiffness comes from the fibers, and the more dense the wood, is, the more fibers there are in a square inch of cross section. There is a correlation between the density of wood and its stiffness (a pretty good one actually), but it's not 100 percent. Balsa is really light, and it's not at all stiff. That correlates. Maple versus roasted maple - the roasted maple is lighter and more stiff - in that case, you're taking out moisture (the fibers are still there, so the stiffness doesn't go down), and you're crystallizing the resin (the glue which holds the fibers together - hardening that "glue" makes the structure stiffer.
Stability is more due to stiffness than weight.
There is a component of stiffness to the long term stability of a piece of wood in respect to resisting the sting torque - the "bridge cables" (heavy flats) on early Fenders, over time, can bend some necks - the 12th fret kink is because those necks were not stiff enough to handle that much torque. So for a good neck you need a certain amount of stiffness.
But....there is, assuming you're using more reasonable strings tension wise, another component in terms of stability that is more important than overall stiffness - the built in stresses that, over time, can relieve themselves. If you cut a piece of wood out of a tree, and let it sit, it will warp very often - there are internal stresses in the wood that are balanced in the whole tree, that when you cut just a section out of a tree, those forces are unbalanced, and they bend the piece of wood. Aging wood (letting it rest after being cut out of the tree until it twists) and THEN making a neck out of it helps, but the species of tree helps a lot as well. Maple is a relatively straight grained wood that is less prone to warping than a lot of other woods - it has less internal stress - the tree "builds" its wood in a manner where, when you cut it, it tends to stay straight. Mahogany (especially Honduras mahogany) is another wood that fits this category - a lot of guitar necks are made from these woods, despite the fact that there are other woods which are definitely stiffer, because the wood itself is less prone to having a lot of internal stresses in it. Roasting maple is a technique that, among other things, relieves the stresses that are there - when you roast maple, you get a VERY stress free piece of wood, and it's very stable.
