The answer to this question is Yes... And No at the same time.
Technically, yes, you can run Matlab without an internet connection. The proviso for this is that you have your own copy of Matlab that you have purchased.
Those who are getting Matlab through UCSB, however, will need to have an internet connection and possibly the use of an OpenVPN connection if you are using Wireless or off-campus to make a connection to the License Server.
The Matlab software UCSB is using is currently is a Site license. UCSB's Site license does require a means to verify the activation of Matlab for a period of time.
UCSB's Site license does not cover a 'stand alone' use. If you need Matlab as a 'stand alone' installation, you will need to purchase a copy of Matlab for yourself to do this.