Suppose that the popularities of everyone are p_1, p_2, through p_20. Then, if you write "~" for "is proportional to", you get a bunch of equations like
0p_1 + 8p_2 + 4p_3 + ... + 4p_20 ~ p_1
7p_2 + 0p_2 + 7p_3 + ... + 9p_20 ~ p_2
...
if you assume that Person #1 is rated 8 by Person #2, 4 by Person #3, and so on, and that nobody rates themselves.
Anyway, you can combine these into a matrix equation
Mv = cv
where M is the matrix with all the popularity ratings that students give each other, v is a vector which says how popular everyone is, and c is a constant. M is known, and you have to solve it for v and c.
Anyway, v is an eigenvector of the matrix M, and finding them is a standard problem.
0p_1 + 8p_2 + 4p_3 + ... + 4p_20 ~ p_1
7p_2 + 0p_2 + 7p_3 + ... + 9p_20 ~ p_2
...
if you assume that Person #1 is rated 8 by Person #2, 4 by Person #3, and so on, and that nobody rates themselves.
Anyway, you can combine these into a matrix equation
Mv = cv
where M is the matrix with all the popularity ratings that students give each other, v is a vector which says how popular everyone is, and c is a constant. M is known, and you have to solve it for v and c.
Anyway, v is an eigenvector of the matrix M, and finding them is a standard problem.
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
The same idea shows up all over the place in linear algebra.