Create a federal law that requires for every sale of a gun:
An underwriting (similar to the insurance industry) which determines someone's suitability to own a gun.
Risk factors would include the buyer's own psychiatric and criminal history, as well as the history of their immediate family and friends.
Firms like Google, Facebook, Twitter, Microsoft and other companies which have access to large amounts of data on the public as well as their connections and relationships with others should work together to help.
These firms could create a joint effort to take the prospective buyer's information and aggregate their analysis of the individual and their "network" into a recommendation.
This recommendation could carry weight in the overall underwriting process.