Speaking as someone who does accept invites from people I don't know, but only if they are sent in the right way, here are my top 5 rules for approaching someone you don't know on LinkedIn.
1. If I don't know you, never just send me an invitation to connect with the standard LinkedIn message. If you do that I have no idea who you are or what you want (and in those circs people usually assume the worst!). I never accept those invites, although if you're lucky I will send you an email asking whether I do in fact know you after all, and you get to have a second go :)
2. Say why you got in contact. It may seem like a silly reason to get in touch but I much prefer a message saying "I got in touch because I saw you on Britain's Brightest" than no explanation at all.
3. Say what it is that we have in common or how we can help each other. Best of all is a message saying "I saw that you have a technology event and I'm interested in that" or even "I see you have really bright software developers, are you interested in interviewing for some more?".
4. Be humorous and sound normal. Cold contacting people is completely usual in the business world. If you sound like that's your view, you'll look more professional to someone reading your message.
5. Never ever lie. I think this is a great rule to live your life by, but it's doubly important when you first meet someone. If you'll lie to me when you first talk to me - can I ever trust you???