Don't rely on connections unless they are in your immediate family but do get a slew of recommendation letters. More importantly, get recommendations from people who are famous in their field or medicine in general. Doctors are people, which is to say, they are stupid enough to believe the opinion of someone they have never met based on the commonly held belief that they are a genius and thus, an excellent judge of someone else's inherent genius, potential, and preparedness. However, celebrity is more important. Hell have a recommendation written by Bono, because he isn't a doctor but everyone knows who he is and his various charitable causes. Also, write a huge check, means more than anything else.