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Concise way to say "increases with n or some term of n"

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Problem Detail: 

I'm writing a thesis proposal and one of the systems involved has unknown complexity. It's not a focus of the proposal, but I wanted to include a line like this, as speculation:

Presumably the complexity of each new connection scales with the number of rules.

Unfortunately I don't know if asymptotic complexity (space and time) will be O($\log{n}$), O($n$), O($n\log{n}$) or even O($n^{99}$). But I do doubt that it will be constant, so increasing $n$ will have some effect on it.

What is a good, accurate way to write this concisely? I'm asking here and not at English Language and Usage because the CS accuracy is essential.

Asked By : David Lord
Answered By : David Richerby

"Presumably the complexity of each new connection is an increasing function of the number of rules."

But you don't mean complexity. Problems have complexity, algorithms have running time (or, more generally, resource usage), functions have growth rate.

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