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In this article:
http://playground.arduino.cc/Main/MAX72XXHardware#Wiring

In particular the circuit of interest is: http://playground.arduino.cc/uploads/Main/MAX72XX_Schematic.jpg

Functionally, capacitors resist changes to voltages. What is the advantage of this in this circuit?

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They are bypass capacitors. As explained in the article:

The capacitors are there to suppress noise signals introduced through the power-supply lines.

The power supply (+5V) should be a pure DC voltage, but noise on the power supply is presented as an AC signal. Capacitors act as short circuits for high frequency signals and open circuits for DC, so the capacitors have no effect on the DC voltage but short the AC noise to ground.

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