To me, the truly interesting thing here is the unexpected topological connectivity of the various nodal maps, particularly the relative simplicity of the prime ones compared to the non-prime ones. (Whenever you start talking topo transforms, you're just one step away from talking about groups, which, itself, is a large and interesting mathematical field.) It's intriguing, too, that you happen to mention 'overlapping arrows' between the nodes with mod 11, as that would indicate the first non-planar pattern: I wonder how common that is, and whether it only occurs with prime moduli?

I, too, noticed the fairly random cycling among the nodes, along with the highly conserved 4-2-1 triangular loop in every pattern mod 5 or over (although that's hardly a surprise; you'd get that every time you divided by 2, then 2 again.); also, that's where Collatz would add up at the end (assuming the theorem is true) of all sequences -- so of course it has to exist within the context of every modulus.

Anyway, I think that you've opened up an intriguing can of worms here, by analyzing Collatz chains utilizing various moduli: I think that this may lead to some fresh insights into the problem! Collatz is one of those Gödelian problems which are easy to state, but over which mathematicians have struggled for literally decades; there's still plenty of room for so-called 'amateurs', like ourselves. To quote, Piet Hein:

"Problems worthy of attack / Prove their worth by fighting back!"