Assign to the $n$ nodes of a path graph vertex weights
forming a permutation of $(0,\ldots,n{-}1)$.
Now iterate the following update repeatedly:
Each node sums the weights of its neighbors, and that node's
weight is replaced (in the next iteration) by the sum $\bmod n$.
Here is the start of an example for $n=4$:
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![Path4][1]][1]
<br />
<sup>
The leftmost node just copies the $1$ of its neighbor.
<br />
The 2nd node from the left is replaced with $(3+0) \bmod 4$.
</sup>
<hr />
Continuing, we fall into a cycle of length $6$:
$$
\left(
\begin{array}{cccc}
 3 & 1 & 0 & 2 \\
 1 & 3 & 3 & 0 \\
 3 & 0 & 3 & 3 \\
 0 & 2 & 3 & 3 \\
 2 & 3 & 1 & 3 \\
 3 & 3 & 2 & 1 \\
 3 & 1 & 0 & 2 \\
\end{array}
\right)
$$
With a different starting permutation, the cycle length can be $3$:
$$
\left(
\begin{array}{cccc}
 2 & 3 & 1 & 0 \\
 3 & 3 & 3 & 1 \\
 3 & 2 & 0 & 3 \\
 2 & 3 & 1 & 0 \\
\end{array}
\right)
$$
Similarly, for $n=5$, cycles of length $2$ and $8$ occur.
But I was surprised to find that for $n=6$, it appears
the process always results in a cycle of length $182$.

> ***Q***. What explains cycles of length $182$ for paths of $6$ nodes?

More generally, what explains the cycle lengths for different $n$?
Here are the cycle lengths I've explored so far, verified by
exhaustive search up to $n=8$. The $n=9$ cycle length of $48$ is prominent
among random permutations but I am not certain there is no other cycle length.
$$
\left(
\begin{array}{cccccccc}
 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9
   \\
 2 & \{1,4\} & \{3,6\} &
   \{2,8\} & 182 & \{6,12\} &
   28 & 48 \\
\end{array}
\right)
$$
I've also explored cycle graphs instead of path graphs.
For a cycle graph of $n=6$ nodes, the process falls into cycles of
length $2$ or $6$.
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![Cyc6][2]][2]
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<sup>
Iteration falls into a cycle of length $2$.
</sup>
<hr />
The process may be similarly defined on any graph.

<hr />
Replying to Moritz Firsching's question in a comment,
a cycle of length $6$ for $n=7$:
$$
\left(
\begin{array}{ccccccc}
 4 & 5 & 6 & 0 & 1 & 2 & 3 \\
 5 & 3 & 5 & 0 & 2 & 4 & 2 \\
 3 & 3 & 3 & 0 & 4 & 4 & 4 \\
 3 & 6 & 3 & 0 & 4 & 1 & 4 \\
 6 & 6 & 6 & 0 & 1 & 1 & 1 \\
 6 & 5 & 6 & 0 & 1 & 2 & 1 \\
 5 & 5 & 5 & 0 & 2 & 2 & 2 \\
 5 & 3 & 5 & 0 & 2 & 4 & 2 \\
\end{array}
\right)
$$


  [1]: https://i.sstatic.net/5nUEd.jpg
  [2]: https://i.sstatic.net/Ctxvq.jpg