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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.

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I don't know if the following results are related, but they might be interesting:

Gitik an I have results, which simply say that (sometimes under the assumption of the existence of large cardinals) one can have a pair (W, V) of models of ZFC, such that adding an $Add(\omega, \kappa)$-generic over V, adds an $Add(\omega, \lambda)$-generic over W, for some $\lambda > \kappa.$

Also, there are results by Shelah and Woodin that one can have a pair (W, V) of models of ZFC, such that W satisfies GCH, V=W[R], for some real R, and In V, the continuum is arbitrary large.

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