For the most part, this won't give you an advantage. All things being equal, you'd probably rather that your app be one of the first apps that they read instead of one of the last. But the effect that this has on your odds of success are much smaller than the effect that your own efforts have on your success. Put another way, I'd rather submit an A+ application right at the deadline than an A- app three weeks early.
For what it's worth, when I went through the process, I submitted my app for each school the same week of the application deadline. I submitted a few days before the deadline, just to avoid any last-minute glitches that could hurt me. But I didn't race to get them in well ahead of the deadline.
P.S. Edit: Applying extra early can be a good way to demonstrate your excitement for a school, but there are no guarantees that it will be read first or that an application reader will even notice the submission date.
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