Well, frankly what you are asking for is that Facebook freeze their entire API and Graph implementation. This simply will not happen. In an ever changing technological world things are going to change. We as developers for a 3rd party platform are responsible for keeping up to date with any changes that 3rd party platform makes.
If you want to keep track of all the changes that Facebook are making to their API's then you should take a look at the Developers Roadmap. They list here all the changes that are being planned.
For serious changes, those what would essentially "break" the current functionality, Facebook guarantees at least a 90 day notification, before the changes are made.
Taken from the Facebook developers roadmap -
In the spirit of openness and transparency and to adhere to our Breaking Change Policy, we publish this roadmap to help developers plan for changes that may require code modifications. Like all roadmaps, it may shift slightly, but we will share insight into what is happening as details become available. We encourage developers to subscribe to our blog, where we announce rollout plans and timing.
Given all of this information, there are still things you can do to make sure your users always have access to your site. One thing would be to provide an alternative login method. It's really useful to be able to hitch a ride on Facebook authentication and seamlessly integrate your site's login with theirs, but what happens if one day Facebook (for some reason) goes down? That would mean that your users would be locked out of your site too! Consider offering an alternative if it is applicable to your situation.