Has there been any research done on the related rates of forcing?

If I force to increase the size of the continuum $\mathfrak{c}$ by 5 $\aleph$'s, say from $\aleph_2$ to $\aleph_7$, how fast does the size of the notion of forcing $\mathbb{P}$ change from the ground model to the forcing extension?

It seems they must just have a constant ratio, but perhaps there are small forcings which have a very big effect.