One of the best known formulas for character development of heroes is Joseph Campbell's monomyth, The Hero with a Thousand Faces.
Campbell identifies this recurring story outline in ancient myths, and many of today's books and movies use a similar structure of a young reluctant hero on a journey, fellow travelers, and an older mentor who is eventually "taken away" so that the young hero must defeat evil alone.
Is there any similar formula that has been used time and again for villains, or anti-heroes?
Some villains are written to be evil just for the same of being evil, which makes them one-dimensional. I am interested in more nuances villains who DO have some character development, and whether there are formulas used in constructing their characters.
I am thinking beyond archetypes and looking for formulas by which villains undergo character development, change, or growth.