Being a fan of LinkedIn for a multitude of reasons, I almost always am willing to connect with other professionals, even if they exist peripherally in my network. To me, that's the benefit of LinkedIn.
I had two experiences recently that to me demonstrate the best and the not-so-best of using LinkedIn to network.
Scenario 1. I had a recruiter contact me about a position at a local company. She is not local and is trying to connect with HR professionals in the area. I share the position through my contacts and respond with some additional (local) resources that an out-of-area recruiter may or may not know. While this took me all of 30 seconds to do, didn't hear a peep. Not a response. Not a thank you. Nothing.
Scenario 2. I had an HR professional from another state connect with me on LI and asked me to share an HR position. My response was the same - shared it broadly with my network of friends and connections and made some additional recommendations to him on great local (free) resources for networking. And received a very nice thank you note.
That's all. If you have someone who does a little above and beyond, say thanks. It's not hard.
Let's make 2011 of gratitude and appreciation.