So this question has been asked four years ago, when websockets weren't supported by many browsers back in 2011 here -> Build a whole site using just websockets (via socket.io and node.js, no Ajax)?
And with the reduction of the size of http headers websockets provide and four years of further development since that question has been asked, would a website build only with websockets help performance and network latency with smaller headers? the initial handshake would occur during a login phase and the connection would timeout after a time period or during logout. So once again, would a website, social network, with real time chat or push notifications benefit from being built solely out of websockets vs http?