For "regular" graphs (lattices, if you will), it's easy to define the percolation threshold $p_c$ as the critical probability beyond which the infinite graph will contain an infinitely large cluster with probability 1. (And below that, with probability 0).
In practice, you can measure that by doing a finite-size analysis and extrapolating.
But what if I'm interested in just the percolation properties of a single instance of a given finite graph?
In that case, I can personally think of a few ways to define a percolation threshold, but it wouldn't be unique. For example, you could say: "For a graph $G$ let $p_c$ be the probability such that the largest cluster will have size $n|G|$ with probability $p > 0.99$ or something like that.
But I wonder if there's one common definition that everyone in the research community uses?
I find it hard to nail down my literature search, probably because I'm using the wrong terms. All papers I could dig up so far talk about finite-size-scaling analysis but with the goal at arriving at $p_c$ for the infinite graph. Pointers in the right direction would be appreciated.