Google matrix

A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages. The PageRank of each page can then be generated iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, irreducible and aperiodic.

Adjacency matrix A and Markov matrix S In order to generate the Google matrix G, we must first generate an adjacency matrix A which represents the relations between pages or nodes. Assuming there are N pages, we can fill out A by doing the following:

A matrix element A i , j {\displaystyle A_{i,j}} is filled with 1 if node j {\displaystyle j} has a link to node i {\displaystyle i} , and 0 otherwise; this is the adjacency matrix of links. A related matrix S corresponding to the transitions in a Markov chain of given network is constructed from A by dividing the elements of column "j" by a number of k j = Σ i = 1 N A i , j {\displaystyle k_{j}=\Sigma _{i=1}^{N}A_{i,j}} where k j {\displaystyle k_{j}} is the total number of outgoing links from node j to all other nodes. The columns having zero matrix elements, corresponding to dangling nodes, are replaced by a constant value 1/N. Such a procedure adds a link from every sink, dangling state a {\displaystyle a} to every other node. Now by the construction the sum of all elements in any column of matrix S is equal to unity. In this way the matrix S is mathematically well defined and it belongs to the class of Markov chains and the class of Perron-Frobenius operators. That makes S suitable for the PageRank algorithm.

Construction of Google matrix G

Then the final Google matrix G can be expressed via S as:

G i j = α S i j + ( 1 − α ) 1 N ( 1 ) {\displaystyle G_{ij}=\alpha S_{ij}+(1-\alpha ){\frac {1}{N}}\;\;\;\;\;\;\;\;\;\;\;(1)}

By the construction the sum of all non-negative elements inside each matrix column is equal to unity. The numerical coefficient α {\displaystyle \alpha } is known as a damping factor. Usually S is a sparse matrix and for modern directed networks it has only about ten nonzero elements in a line or column, thus only about 10N multiplications are needed to multiply a vector by matrix G.

Examples of Google matrix An example of the matrix S {\displaystyle S} construction via Eq.(1) within a simple network is given in the article CheiRank. For the actual matrix, Google uses a damping factor α {\displaystyle \alpha } around 0.85. The term ( 1 − α ) {\displaystyle (1-\alpha )} gives a surfer probability to jump randomly on any page. The matrix G {\displaystyle G} belongs to the class of Perron-Frobenius operators of Markov chains. The examples of Google matrix structure are shown in Fig.1 for Wikipedia articles hyperlink network in 2009 at small scale and in Fig.2 for University of Cambridge network in 2006 at large scale.

Spectrum and eigenstates of G matrix

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