I don't begrudge anyone doing what they have to do if they really need a job - but the idea that you're a LinkedIn connection away from the job is pure fantasy. If it's on LinkedIn it's on other job boards, and the only "connections" you'll make inherit to the site are recruiters who are looking to fill positions with people who meet the minimum qualifications for jobs that just need to be filled - not anyone scouring for the right hire.
LinkedIn is the Facebook of jobseekers. I mean that in both user interface and what you should expect to get out of it. If you don't keep up with someone other than by FB posts - you're probably not really very good friends; if you can't keep up up with someone professionally beyond LinkedIn they're probably not really a connection. When you leave a job go with the email and phone number (personal if appropriate) of your bosses you respect and peers you assess as potential movers and shakers. Find a reason to stay in touch. Ask them their opinions of developments in your industry and add some personal regards.
Everyone knows LinkedIn is just to try to get a hand in your career. While everyone who is smart knows your stay in touch plan is the same, just by the nature of it if you actually put a little effort in with people you genuinely respect it can't help but turn into a real connection. They'll be much more likely to give you real help or advice when you need it rather than just "liking" your fluff post and referring you to some job you could have found on any job board.
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