The user making the connection won't show up in the JFBConnect user map area if you're logging in with LinkedIn. That will still be in the JLinked User Map area, for now. Can you check there to see if it's working correctly?
If it's not showing there *and* the EasySocial profile plugin is disabled, that means the issue isn't related to that plugin. The problem is something specific to JLinked/LinkedIn.
In that case, can you clear browser cache, disable Joomla's caching and disable any SEF extensions you have enabled and then try again? That will narrow a lot of possible issues really quickly. If that works, we can narrow down things further, if not, it means that it's likely a communication issue with LinkedIn itself (either the API/Secret key are incorrect or LinkedIn is responding incorrectly).
Let me know how the above test goes, and we'll move on from there.
Thanks,
Alex