I’m not exactly sure if I understand how these weights that are assigned to different vertices are supposed to be used by the algorithm. Also how are these numbers calculated again? The previous slide stated that $u=3/16$ if the degree is 3, but here the weights of the surrounding vertices are $1/16$.

This vertex has valence 6, not 3.

From my understanding, the weights assigned to neighboring vertices are 3/48 = 3/(8n) where n = 6 because the center vertex's degree is 6.

I'm confused how these weights will change the vertex. Isn't there symmetry here?

If the vertices in 3D space were perfectly symmetric about the center point and all in the same plane, then yes the point would not move. But in general, the vertices will not be symmetric and not be in the same plane.

ah I see, missed that

As Sean said u = 3/16 if n == 3 else 3/(8*n) is a nice way to see why we have $\frac1{16}$ here