In the games, Minato can gain experience boosts for his Personas through his Social Links, bonds he has with friends. The higher the Link Rank, the greater the boost. Upon maxing the S.Link, he gains access to the ultimate Persona of that Link. Each Social Link corresponds to a specific Arcana, and each Arcana can only have one Social Link. As far as his links back home go, Minato's Social Links are as follows:
0. The Fool: Specialized Extracurricular Execution Squad [Rank MAX]
I. The Magician: Tomochika Kenji [Rank MAX]
II. The Priestess: Yamagishi Fuuka [Rank 5]
III. The Empress: Kirijou Mitsuru [Rank 4]
IV. The Emperor: Odagiri Hidetoshi [Rank MAX]
V. The Hierophant: Bunkichi and Mitsuko [Rank MAX]
VI. The Lovers: Takeba Yukari [Rank 5]
VII. The Chariot: Miyamoto Kazushi [Rank MAX]
VIII. Justice: Fushimi Chihiro [Rank 3]
IX. The Hermit: Y-ko/MAYA [Rank MAX]
X. Fortune: Hiraga Keisuke [Rank MAX]
XI. Strength: Nishiwaki Yuko [Rank 4]
XII. The Hermit: Maiko [Rank MAX]
XIII. Death: Pharos [Rank MAX]
XIV. Temperance: Andre Roland Jean Gérard ("Bebe") [Rank MAX]
XV. The Devil: Tanaka [Rank MAX]
XVI. The Tower: Mutatsu [Rank MAX]
XVII. The Star: Hayase Mamoru [Rank MAX]
XVIII. The Moon: Suemitsu Nozomi [Rank MAX]
XIX. The Sun: Kamiki Akinari [Rank MAX]
XX. Aeon: Aigis [Rank 7]
XX. Judgment: Nyx Annihilation Team [Rank MAX]
For now, that's how it is. This list proooobably isn't liable to change anytime soon.
Well, if anyone wants to start an S.Link with Minato in camp, just say so and we can probably work something out? They won't replace the Links he already has, but upon starting the link in question, it will effectively overrule his home-Links so long as he and the character he's sharing a link with are both in camp. For now, I'm gonna go ahead and say that Fool, Death and Judgment are off-limits, however.