Understanding Google's PageRank Algorithm through Eigenvectors and Determinants, Slides of Engineering Mathematics

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Google and eigenvectors

Last week

Operation Effect on the determinant

Transpose No change

Exchange two rows/columns Change the sign of determinant Multiply a row/column by a constant k

multiply the determinant by k

Add a constant multiple of a row/column to another row/column

no change

The followings are equivalent

( A is an nxn matrix)

• There is a nonzero solution to A x = 0.
• The columns of A are linearly dependent.
• The rows of A are linearly dependent.
• det( A ) = 0.
• The inverse of A does not exist.

GOOGLE’S PAGERANK ALGORITHM

The Pagerank algorithm

• Larry Page and Sergey Brin, the founders of Google, invented the pagerank algorithm for ranking webpages.
• Ideas: the importance of a page is high if
  • it is linked from many other places.
  • it is linked from a page which is known to be important.
• The underlying mathematics of the Pagerank algorithm is the calculation of eigenvector. Pagerank algorithm

A Toy Example

• A toy example of four web sites.
• An arrow from page A to page B means that there is a link from A to B, and so on. - Page A has three out-going links - Page B has two out-going links - Page C has one out-going link - Page D has one out-going link

A B

C (^) D http://williamcotton.com/pagerank-explained-with-javascript

Repeat many times …

A B

C (^) D

Step 1

1.333 0.

1.5 (^) A B

C (^) D

Step 2

1.333 0.

A B

C (^) D

Step 3

1.167 0.

1.583 A^ B

C (^) D

Step 4

1.278 0.

Matrix representation

• Let the numbers associated with each page be represented by the components of a 4x vector.
• Let

A B C D A B C D

A B

C (^) D

xA xB

xC xD

If it converges, the limiting values

can be found by solving equations

where k is any real constant (^) when the sum is 4

Important Remarks

• x is a solution to ( M – I ) x = 0.
• If x is a solution, then any scalar multiple of x is also a solution. - Mx = x  M (k x ) = k x for any real number k
• Mx = x has a non-zero solution, because the determinant of M – I is 0. - We can see this by noticing that the four rows of M – I sum to the zero vector. Hence, the rows are linearly dependent, which implies that det( M – I )=0.

Eigenvalue and Eigenvector

• Given a square matrix A , a non-zero vector v is called an eigenvector of A , if we an find a real number  (which may equal to zero), such that
• This number  is called an eigenvalue of A , corresponding to the eigenvector v.

Matrix-vector product Scalar product of a vector

Important notes

• If v is an eigenvector of A with eigenvalue , then any non-zero scalar multiple of v also satisfies the definition of eigenvector.
• The PageRank algorithm is designed so that the largest eigenvalue is =1, and the algortihm output an eigenvector corresponding to =1.

k  0

Invariant direction

• An eigenvector points at a direction which is invariant under the linear transformation induced by the matrix.
• The eigenvalue is interpreted as the magnification factor.
• L ( x,y ) = (x+2y, 3x-4y)
• If input is (2,1), output is magnified by a factor of 2, i.e., the eigenvalue is 2.

The direction is the same as the input vector.

Another invariant direction

• L ( x,y ) = (x+2y, 3x-4y)
• If input is (-1/3,1), output is (5/3,-5). The length is increased by a factor of 5, and the direction is reversed. The corresponding eigenvalue is -5.

The output vector is in the reverse direction of the input vector.