There really is no algorithm for this, instead it's a simple member count prerequisite approach.
Here are the member numbers for each tribe level:

Level 1 - 0+ members
Level 2 - 9+ members
Level 3 - 40+ members
Level 4 - 100+ members
Level 5 - 300+ members
Level 6 - 1,000+ members
Level 7 - 3,000+ members
Level 8 - 10,000+ members
Level 9 - 30,000+ members
Level 10 - 100,000+ members
Level 11 - 300,000+ members
Level 12 - 1,000,000+ members (currently maximum level)