see vs run across

see is a noun but run across is not a noun.

Word	Noun	Adjective	Verb	Adverb
see	Yes	No	Yes	No
run across	No	No	Yes	No

As verbs, see and run across are synonyms defined as:

Other synonyms of see include come across, encounter, meet, run into.

see (noun)	run across (noun)
the seat within a bishop's diocese where his cathedral is located

see (verb)	run across (verb)
deem to be	come together
perceive (an idea or situation) mentally
get to know or become aware of, usually accidentally
be careful or certain to do something; make certain of something
deliberate or decide
make sense of; assign a meaning to
find out, learn, or determine with certainty, usually by making an inquiry or other effort
match or meet
receive as a specified guest
imagine; conceive of; see in one's mind
come together
conduct someone someplace
perceive by sight or have the power to perceive by sight
perceive or be contemporaneous with
see or watch
see and understand, have a good eye
observe as if with an eye
undergo or live through a difficult experience
observe, check out, and look over carefully or inspect
date regularly; have a steady relationship with
go to see a place, as for entertainment
go to see for professional or business reasons
go to see for a social visit
take charge of or deal with