*Options and Futures* ` ` ` `*A brief note on topology*

## A brief note on topology.

The cube (or hypercube or n-dimensional cube) architecture is
flexible, easy to remember, and has a cute name, but it is not the
cheapest or fastest way to connect a set of nodes.
Let's look at the simplest three-dimensional
cube.

It connects eight nodes. ` `
Every node is connected to three other ones (in the x, y, and z
dimension). ` `
Starting from any one node, I can reach one in 0 hops (the one
I started with, let's say **a**),
three others in 1 hop (**b**, **g**, **d**),
three more in the 2nd hop (**h**, **f**, **c**),
and the remaining eigth node in a 3rd hop (**e**);
taking 1 1/2 hops to
reach any node on average, and 3 in the worst case.
But the Petersen Graph

connects more nodes with a better worst-case behavior: reaching
one after 0 hops (take, again, **a**), three after 1 hop (**h**,
**b**, **e**), and six more after 2 hops (**c**, **d**,
**f**, **g**, **i**, **j**); ten in total. ` `
Again, the average hop-count is 1 1/2, but the worst case is only 2.

Does this scale? ` `It doesn't scale to the general case,
but to which *does* it scale?