a politically 'connected' businessman [=a businessman who has relationships with people who have political power]
In my mind, the two places 'are connected'. [=I think of the two places as being related to each other]
They 'are' somehow 'connected' to the royal family. [=they are related in some way to the royal family]
(..) ... and without all the trains. King's Cross, is that right? This is, as they say, 'your' party. I expect you now realize that you and Voldemort... ... have been connected by something other than fate. Since that night in Godric's Hollow, all those years ago. So it's true then, sir. (..)2011 Harry Potter and the Deathly Hallows: Part 2