advantages in applying early to PhD programs
The only likely advantage to applying early is that you won't run into unexpected events that make it difficult to meet the deadline, and that this could improve the quality of your application (especially the written/essay portions) as well as avoid simply failing to meet the deadline at all.

That said, this can be a very valuable advantage. You'll find a lot of other questions here from people panicking about having missed application deadlines, having deadlines approaching too quickly, being unable to reach their writers for recommendation letters, etc.

I would not recommend applying extremely early (as-in, months in advance) - in that case, you'll instead be regretting having missed your most recent accomplishments. Give yourself some cushion against the deadline, and don't worry about gaming the system beyond that. No one cares (or probably even knows) if your application is officially submitted 0.01 or 1 or 2 or 3 weeks before the deadline: the deadline is there so that the admissions committee has a known day to start the job of reviewing while being fair to everyone. That's the only game.