Roles are generally a set of capabilities assigned to a group, not specific users. See this page for more.
I don’t really think you want to give those who register Contributor right as you’ll see in the chart I linked to will give them the capability to edit and delete posts. If you do not want that, you should have them set as the subscriber role.
The capabilities for Connections works the same way as those for WordPress. If a role is assigned a capability, then anyone that is assigned that role can perform that action.
If you want registered users to be able to maintain their own entry in Connections, then you need Link.
I hope that helps.