I have to choose between the above mentioned frameworks for developing a big social networking Web site. Having used Zend framework for several projects before, I am biased towards using it, however, I would like to have a fair opinion.
We are mainly looking for performance and extensibility.
Use what you're comfortable with. That's the most important thing... There's no sense using a tool you don't understand since you won't be nearly as productive (among other issues)...
In my opinion there is no right or wrong in this case. You should choose the framework you can develop best yourself. The better you know, the better and more performant your application will be. It is possible to write fast and slow with every framework. So in your case it is absolutly ok to choose Zend. Keep in mind that most of the framework have their own benchmark where it is praised as the best performing solution ;)
Use Zend Framework then. Extensibility is a given with Zend Framework and as for performance well it's up to how you code. Zend Framework is more of a VC you're the one that implementing the M. So going with the Fat Model and skinny Controller programming logic, you should be able to make Zend Framework decently quick? I'm sure you can do lazy loading in there and get close to as fast as Yii (since Yii does lots of lazy loading).
No idea what CI is.
Yii is much faster to develop just fyi.