Connecting with someone
Connect signals interest; when it is returned you are matched, and matches show your mutual connections.
Connecting is one-sided until it is not. You ask; if they ask back, you are matched, and the match shows on both profiles along with any connections you have in common.
You do not have to be connected in order to message someone — a first message from a stranger simply asks them to allow it. Connecting is the durable version: it lasts, it shows mutuals, and it feeds what the map recommends.