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:
[![K3][1]][1]
The $3$-path starting with $(1,0,0)$ goes in three steps to $(0,0,0)$ and then is stable:
[![P3][2]][2]
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:
[![Y][3]][3]
(If animated, the graph would "blink.")
Q. Which graphs have blinking assignments?
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:
[![BlinkWheels][4]][4]
(Bit assignment thanks to Tony Huynh.)