For each arithmetic function $f:\mathbb{N}\rightarrow \mathbb{N}$ and each $n\in \mathbb{N}$ you can define a relation $f_{\textsf{mod } n}:[n]\times[n] \rightarrow \{0,1\}$ with

$$f_{\textsf{mod } n}(i,j) = 1 \iff f(i) \equiv j \textsf{ mod } n $$

for $[n] = \{0,1,\dots,n-1\}$.

Plotting this relation (the "modular graph") shows highly regular patterns for simple functions like addition $k \mapsto k + c$ or multiplication $k \mapsto c \times k$. To the left the graph with all nodes arranged on a circle, to the right the adjacency matrix. (If you want you can check it all here.)

But the modular graphs for multiplication look more diverse, intricate and somehow more "random" than the graphs for addition. The modular graphs for squaring $k \mapsto k^2$ look even more "random" and have no apparent symmetry:

But to my surprise, the modular graphs for higher powers look more regular again:

Note that the modular graphs for the third and the fifth power have at least one symmetry even when $n$ is chosen to be a prime number:

My question is two-fold:

How do I understand the exceptional status of squaring with respect to the "randomness" of the modular graphs?

Are there genuinely different sequences of operations that yield ever more random graphs?