For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a *bit assignment* for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the sum of the bits of $x$'s
neighbors is odd, and $0$ if that sum is even.
For example, the $3$-cycle $K_3$ with bits $(1,0,0)$ goes to $(0,1,1)$ and then is stable:

The $3$-path starting with $(1,0,0)$ goes in three steps to $(0,0,0)$ and then is stable:

Say that $G$ has a

*blinking assignment*if there is a bit assignment that flips to its complement in one step, and then returns to the original assigment, forming a cycle of length two:

(If animated, the graph would "blink.")

. Which graphs have blinking assignments?Q

For example, $K_3$ has no blinking assignment, but the star $S_k$ for $k$ odd does. Many other questions could be asked (e.g., concerning longer cycles), but I'll focus on the above for now.

Because the update rule is to replace a node's weight with the sum of its neighbors' weights $\bmod 2$, it seems possible this process has been studied for some $\mathbb{Z}_n$. If so, I would appreciate a pointer.

*Added*animation just for fun:

^{ (Bit assignment thanks to Tony Huynh.) }