Previously, Abhijeet has highlighted three awesome sites where you can create your own social network easily. However, if you are one of those who wants more control of the site, including the branding, the domain name, the features, the design, the community rule, or even the database, the best way is to install and run a social networking software on your server.
While there are plenty of scripts (both free and paid) that you can use to run your own social network, there are only a few that are stable, good and easy to use.