Divisibility as a walk on a graph

This graph lets you find out whether a given number is divisible by 7 or not. The challenge for you is to figure out WHY this works. Here is the method:

Write down a number n. Start at the small white node at the bottom of the graph. For each digit d in n, follow d black arrows in a succession, and as you move from one digit to the next, follow 1 white arrow.

For example, if n = 325, follow 3 black arrows, then 1 white arrow, then 2 black arrows, then 1 white arrow, and finally 5 black arrows.

If you end up back at the white node, n is divisible by 7.