Shweiki Media and Avital Eidenbom from Studio 5 Innovation have put out this short webinar about using Facebook Insights and Graph search to get more out of your social media content.
One Graph Search tip from the video is to search for other pages liked by your fans, using the "Pages liked by people who like YourBrandHere" query. Look at the pages that it returns, and look at the kinds of posts they have made with high engagement.
Facebook recently launched a Pages to Watch feature, which lets you keep an eye on competitors' pages as well.
Image via YouTube