I wish to create an application which me and my friends would subscribe to and which in turn can monitor a count of items in the news feed using the required permissions (kind of a social network statistics).
I read the developer documentation, but I couldn't find out if it's possible. Subscribing to my application (and allowing the permissions) is a one-time process for my friends. From then on, can my application read the fields mentioned in Graph API for the subscribed users at any time (without the user's involvement)?
You might be interested in using the Real-Time Updates API from Facebook. Using this, you will no longer have to continuously "poll" the Graph API. Rather, Facebook will send the updates to your server.
Here are some sample implementations of the API, https://github.com/facebook/real-time/tree/master/samples.