• Reconnect with people you know first. Your best contacts are the people who already know you. Old classmates. Professors. Former colleagues. Where are they now? Maybe they're in positions to help you.
• Unless your contact is a recruiter, don't treat them like one. If someone agrees to connect with you, don't follow up with a resume, unless they request it. Not everyone is in a position to help you get a job. Don't pressure them.
• Give them reason to want to know you. You approached this contact for a reason: They are knowledgeable... They are well-connected... If they accept your invitation, turn the tables and get them to want to know you. They've given you valuable information. Give them something valuable in return. If you want a stranger to ever refer you for a position, become a connection worth keeping. 
• Be patient. If you've connected with busy or well-known individuals, they are not spending the day responding to every email inquiry. Some may eventually get back to you, but it may take a while. Most may not contact you at all. Still they are resources worth keeping. Don't take it personally.
• Don't become a stalker. If you find yourself on the verge of sending angry emails or rants about someone who hasn't responded to you, you're about to cross the line. Again, don't take it personally.
• Know when you're being treated like a number and accept it. Some people just like building buddy lists. To them, it looks impressive to have 2000+ connections attached to your name,  yet you have no real connection with them. Again, don't let that stop you from following their work if it is beneficial to you.

Before all of this social media, we didn't call everyone who gave us a business card. In life, we only connect with a small percentage of the people we meet on- and off-line. So don't get frustrated now that the world is a lot more accessible. Your don't have to be buddy-buddy to benefit from the connections you make.