There could be only one!
That’s the mentality of social networks, a social network is by definition supposed to connect people and due to the fact that often people travel, change their living place or even migrate, a real social network can not exist on a local scale not even on a national scale.
Facebook is by far the biggest and most powerful social network, but in a way it sees twitter as a threat, because people use Twitter to connect and share info with strangers while they use Facebook to connect with friends, facebook limited the number of friends to 5000 to reduce abuse, thus twitter is the favorite platform for many celebrities to reach to their fan base.
Facebook war on twitter started with the introduction of the subscribe button that allows you to see the public posts of people you subscribe to, but now there are strong rumors that Facebook will change the name to Follow, making it even more obvious, and probably slowing down twitter growth worldwide.
Another step facebook taking against twitter is a proxy war through Instagram, that was acquired recently by Facebook, Instagram disabled the ability for Twitter to properly display Instagram photos on its Web site and in its applications.