(I'm pretty sure the answer is going to be "the phone will work on any network" but seeking confirmation)
This phone is labeled as a 2Degrees phone but they aren't locked, I got one for my wife for Xmas and she's on Vodafone, the phone, her SIM and her connection are working fine.
My brother-in-law is keen for one now for his wife (she's on Telecom) and was told by DSE that "they won't work that well with a Telecom SIM"
http://www.dicksmith.co.nz/mobile-phones/samsung-galaxy-siii-mini-white-2degrees-dsnz-em0110
Interesting on their site that the 'Telecom' version of this phone is $50-60 more expensive.