Wolfram Alpha Blog : Yes, That’s Dumb—A Lexicographic Footnote

An unbounded number of networks would be made of an unbounded number of nodes in the minimal case, and I have a hard time describing the number of English words as "unbounded". If you apply the full power of Internet debate (now with extra axioms!) to the twisting and spinning of the term and streeeeetch the meanings as hard as you can, maybe you can get there, but not by any sane method.

It is possible that "black" and "white" are in the same synonym network, perhaps linked by words like "ash". That would be amusing. Someone can check it on WolframAlpha.

The networks are "non-overlapping", because of the definition of a synonym network. You start with a seed word, and grow it until it stops growing. The English language is finite, so K must be finite. The example network containing "black" is large, so K is expected to be small, (unless there are a few large networks, and many tiny ones).

Finding K is not difficult at all. Pick a word, "black". Create the synonym network containing "black"and all words linked to it through the "synonym" relationship. Mark those words (roughly 26,000 of them) as "covered". K = 1.

Pick any other word not already covered. Grow the next synonym network. Mark those words covered. K = K + 1

Repeat until there are no words left.

K is the number of synonym networks.