2024-12-16

In the movie recommendation application we want to modify the process to now incorporate the actual ratings of movies. Now when comparing two covers $U$ and $V$ there are two variables of interest: rating of movie $u$ and rating of movie $v$. While it makes sense to recommend movie $u$ if it has a high rating, the rating for movie $v$ may be low making it a worse recommendation.

For that reason, instead of making an undirected graph, we make a directed graph where if two cover elements $U$ and $V$ intersect than we create two arcs $(u,v)$ and $(v,u)$.

We want to apply weights to the arcs were a smaller number implies higher rated. I think the following modification to Steinhaus distance is worth considering:

$$d(U,V)=1-\frac{1}{|U\cup V|}\sum _{i\in U\cap V}{r}_i$$

where $r_i\in [0,1]$ is the normalized rating user $i$ ranked movie $v$. We apply the weight $d(U,V)$ to arc $(u,v)$ and the weight $d(V,U)$ to arc $(v,u)$.

The stable path finding algorithm requires no modifications under these changes.

This new “distance function” now recommends a movie $v$ based on how many users who saw both movies $u$ and $v$ rated movie $v$. In the arc $(u,v)$ we now care if users liked movie $v$ but do not care about their opinion on movie $u$. Those opinions give influence to arc $(v,u)$.

Under this new design the Mulan to Moulin Rouge route becomes:

1. Mulan (1998)
2. Tarzan (1999)
3. Winnie the Pooh and the Blustery Day (1968)
4. Charlotte’s Web (1973)
5. Jungle Book, The (1967)
6. American Tail, An (1986)
7. Money Pit, The (1986)
8. Blind Date (1987)
9. Let It Ride (1989)
10. Mysterious Skin (2004)
11. Brokeback Mountain (2005)
12. Moulin Rouge (2001)