Would a payment propogate through the node quickly enough to make it practical for something like this?

That is a good question. How fast do payments propagate? Has anybody ever timed a payment?

And better yet, is there a way to trace through the nodes from endpoint to endpoint to see how many "jumps" a transaction has to make before it reaches from one end to another?

It would be something like X = N/N/(8^X)... I think. Where X = number of hops and N = number of nodes.

So say we have 50,000 people with nodes running.

X = 50,000/(8^X)

X(8^X) = 50,000

Uh...

It's been a while since I took a math class.

The shortest number of paths would be approximately log( N )/log( 8 ). So about 5.2 hops for 50,000 nodes of which each are connected to 8 non-repeating nodes. I talk to 8, they talk to 64, they talk to 512, they talk to 4096, and they talk to 32,768, and 20% of them talk to the rest.

But with a mesh network, there would be a lot of overlap (I talk to a peer, who talks to a peer, who talks back to me, for instance). There's probably a formula, and it's probabilistic, probably.

I'd guess it's would average around 2x the above figure, so 10.4 hops. to reach everybody, on average, in a 50,000 node mesh network.

PS, the math is:

8 ^ X = N

log ( 8 ^ X ) = log ( N )

X log ( 8 ) = log ( N )

X = log ( N ) / log ( 8 )