"any" versus "arbitrary"

Which of the following sentences are correct?

The relation between any two nodes is associated with a real number that indicates the probability.

The relation between two arbitrary nodes is associated with a real number that indicates the probability.

The relation between two nodes is associated with a real number that indicates the probability.

• All of them parse well, though the last one seems to indicate the use of "any" and "arbitrary" as a redundant, and therefore unnecessary, term. This seems like proofreading, though. You may want to edit your question to not present that vibe. Aug 8, 2014 at 6:27

While the sentences are grammatically correct, they are backwards in that they emphasize association of two nodes with a number. That may be what you mean; but more probably, the sentences should say a number is associated with any pair of nodes. For example:

Each pair of nodes has a probability assigned to it.

If you need to emphasize that a real-number datatype is used for representing probability values, add real-number before probability in the above; otherwise, leave it out, as it's just clutter.

As @SrJoven has commented, they're all "correct" grammatically.

So the question is, which is correct for your intended meaning?

If the first sentence (any) conveys your intended meaning correctly, then the latter two are inferior. This is because they both imply qualification in some way of which nodes this can be said about. (Either they must be arbitrary nodes, whatever that could mean in a given context, or in the other case, the statement might be true about two particular nodes and not other nodes.)

• Use of arbitrary isn't inferior to any; both have the same meaning in context of graph theory, with arbitrary merely more explicitly saying the two nodes referred to were not specially chosen Aug 8, 2014 at 13:07
• @jwpat7 the OP does not mention graph theory. Aug 8, 2014 at 13:26