Chapter 8 Google Matrix Flashcards
What does it mean for the row eigenvectors of H is 1 is a simple eigenvalue of H
If we calculate any row eigenvector w^ tranpose of H where all elements are positive then all other row eigenvectors will be a multiple of W^tranpose
What are problems with using W^tranpose, row eigenvector of H to find the page rank vector?
H may have 1 as a repeated eigenvalue so this doesnt hold.
Calculating the row eigenvector of H w^tranpose can take too long
Also method of squaring H to find w ^tranpose can take too long depending on the speed of convergence
What are the properties of the google matrix
- It’s all positive
- Its stochastic
- Eigenvalues of G and cH are the same except the perron eigenvalue c of cH corresponding to the perron eigenvector e is replaced by 1
- If x is a eigenvalue of G different from 1 then |x|<=c
What do we call the real number c as part of the google matrix?
Damping factor
What does c being close to 1 mean in the google matrix?
G will be close to H and ranking is close to the original Page Rank using H. c close to 1 means G converges slowly
What does c being close to 0 mean in the google matrix?
G will be not very close to H and ranking is different to the original Page Rank using H. c close to 0 means G converges quickly
What does choice of C involve compromise on?
Finding a balance for the application between the H influence and speed of convergence (meaning getting a good approx of w ^transpose fast
What are the advantages of Google matrix?
Its positive and has simple perron eigenvalue 1
There is only one vector w^tranpose with w^tranpose x G= w^tranpose can be deduced from the limit of G
C can be adjusted to the application
Even if limit of H exists the convergence can be very slow.
What are the disadvantages of changing to Google matrix
PageRank using H is only based on principle that important pages are pointed to by many pages in particular many important pages.
G is not as sensitive to the properties of the web as H
In a sports context what does an arrow pointing from team i to team j mean?
Indicates team i lost to team j
How can we have a weighted adjacency matrix in sport?
Weight on each pointer can be the positive difference between the teams
What does Aij mean in the sport teams weighted adjacency matrix?
Aij means team i lost to team j by aij points
What should be done if when converting and adjacency matric to a H matrix there is a row of all zeros?
Assume even probabilities across the row.
