She attended the fund-raiser 'at the invitation of' the committee chairperson. [=she attended because she was invited by the committee chairperson]
You have a 'standing/open invitation' [=you are always welcome; you are always invited] to visit us any time.
The event is 'by invitation only'. [=only people who have been invited to the event can attend it]
(..) And what things? Thank you. - Okay, I'm gonna go find my family, who are impatiently waiting to embarrass me. Bye. - Visit us soon yeah? - Soon as I get an invitation. So, how long you two known each other? A few weeks now, we actually met when Anastasia interviewed me for the student newspaper. (..)2015 Fifty Shades of Grey