At their maximum separation, Mars & Earth are about 20 minutes apart. If we had 10 satellite data centers all in perfect alignment (disregarding the sun, which obviously makes a hash out of things) they'd still each be 2 minutes apart.

Once you take into consideration the sun, plus the fact that the you'd need to cover the full disk to keep all data centers within a few minutes of another one in an unbroken chain back to both planets, I just don't get the math involved here.

But, I'm also terrible at both math and visualization, so I readily concede I may be missing something obvious.

Think of it more like 3 circles.

The inner circle has Earth's orbit in it. The outer circle is Mar's orbit.

The middle circle would be a ring of relatively stationary satellites in between them.

And in the center of all 3 circles is the Sun, which will not allow radio signals to pass through.

I drew a crappy illustration to demonstrate: https://ibb.co/tP2rkzS0

When Mars and the Earth are on opposite sides of the sun, a satellite ring can transmit around the sun and keep the communication lines open.

Having a ring of relay satellites gives you a set distance to transmit from Mars. The satellites can then transmit their received data from the one that is closest to Mars to the one that is closest to Earth, which would then send the data to Earth.

This is helpful for a variety of reasons, but the most important one is that with this setup, even when the Sun is in between Earth and Mars, you could still send data around the sun.

Constant communication, no communications breakdowns. Even if 1 satellite failed for some reason, a bit of maneuvering would allow the others to backfill the gap until it could be repaired or replaced.

Even when Earth and Mars are close together, it would still be smart to use the relay so that the power levels are easily calculated and maintained.