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.

  • $\begingroup$ there is some work on the emergence of a giant component that seems related math.cmu.edu/~af1p/Texfiles/quasigiant.pdf $\endgroup$ Oct 17, 2016 at 20:01
  • $\begingroup$ A nice find, but they still focus on the limit $n \rightarrow \infty$ from what I could skim $\endgroup$
    – Lagerbaer
    Oct 17, 2016 at 20:19
  • $\begingroup$ I think you will be hard pressed to find literature for the truly finitary case, (no limits and no asymptotics) exactly because there is no way to make a canonical choice among various possible definitions. $\endgroup$ Oct 17, 2016 at 21:05
  • $\begingroup$ Okay. Well, I guess that's cool too. Just wanted to make sure that when I use my definition I'm not wasting time by not following the "official" definition :) $\endgroup$
    – Lagerbaer
    Oct 17, 2016 at 23:20


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.