Decoherence-free subspaces

Fig.1. Google matrix of Wikipedia articles network, written in the bases of PageRank index; fragment of top 200 X 200 matrix elements is shown, total size N=3282257 (from [19])

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 rank of each page can 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:

1. A matrix element $A_{i,j}$ is filled with 1 if node $j$ has a link to node $i$, and 0 otherwise; this is the adjacency matrix of links.
2. 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 $k_{j}$ where $k_{j}$ 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 $a$ to every other node.
3. 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

Fig.2. Google matrix of Cambridge University network (2006), coarse-grained matrix elements are written in the bases of PageRank index, total size N=212710 is shown (from [19])

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

$G_{ij}=\alpha S_{ij}+(1-\alpha ){\frac {1}{N}}\;\;\;\;\;\;\;\;\;\;\;(1)$

By the construction the sum of all non-negative elements inside each matrix column is equal to unity. The numerical coefficient $\alpha$ 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[1,2].

Examples of Google matrix

An example of the matrix $S$ construction via Eq.(1) within a simple network is given in the article CheiRank.

For the actual matrix, Google uses a damping factor $\alpha$ around 0.85 [1,2,3]. The term $(1-\alpha )$ gives a surfer probability to jump randomly on any page. The matrix $G$ belongs to the class of Perron-Frobenius operators of Markov chains [1]. 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

File:Googlematrixcambridge2006spectrum.gif

Fig3. The spectrum of eigenvalues of the Google matrix of University of Cambridge from Fig.2 at $\alpha =1$, blue points show eigenvalues of isolated subspaces, red points show eigenvalues of core component (from [14])

For $0<\alpha <1$ there is only one maximal eigenvalue $\lambda =1$ with the corresponding right eigenvector which has non-negative elements $P_{i}$ which can be viewed as stationary probability distribution [1]. These probabilities ordered by their decreasing values give the PageRank vector $P_{i}$ with the RageRank $K_{i}$ used by Google search to rank webpages. Usually one has for the World Wide Web that $P\propto 1/K^{\beta }$ with $\beta \approx 0.9$. The number of nodes with a given PageRank value scales as $N_{P}\propto 1/P^{\nu }$ with the exponent $\nu =1+1/\beta \approx 2.1$ [4,5]. The left eigenvector at $\lambda =1$ has constant matrix elements. With $0<\alpha$ all eigenvalues move as $\lambda _{i}\rightarrow \alpha \lambda _{i}$ except the maximal eigenvalue $\lambda =1$, which remains unchanged [1]. The PageRank vector varies with $\alpha$ but other eigenvectors with $\lambda _{i}<1$ remain unchanged due to their orthogonality to the constant left vector at $\lambda =1$. The gap between $\lambda =1$ and other eigenvalue is $1-\alpha \approx 0.15$ gives a rapid convergence of a random initial vector to the PageRank approximately after 50 multiplications on $G$ matrix.

Fig4. Distribution of eigenvalues $\lambda _{i}$ of Google matrices in the complex plane at $\alpha =1$ for dictionary networks: Roget (A, N=1022), ODLIS (B, N=2909) and FOLDOC (C, N=13356); UK university WWW networks: University of Wales (Cardiff) (D, N=2778), Birmingham City University (E, N=10631), Keele University (Staffordshire) (F, N=11437), Nottingham Trent University (G, N=12660), Liverpool John Moores University (H, N=13578)(data for universities are for 2002) (from [7])

At $\alpha =1$ the matrix $G$ has generally many degenerate eigenvalues $\lambda =1$ (see e.g. [6,7]). Examples of the eigenvalue spectrum of the Google matrix of various directed networks is shown in Fig.3 from [14] and Fig.4 from [7].

The Google matrix can be also constructed for the Ulam networks generated by the Ulam method [8] for dynamical maps. The spectral properties of such matrices are discussed in [9,10,11,12,13,14,15,16]. In a number of cases the spectrum is described by the fractal Weyl law [10,12].

Fig5. Distribution of eigenvalues $\lambda$ in the complex plane for the Google matrix $G$ of the Linux Kernel version 2.6.32 with matrix size $N=285509$ at $\alpha =0.85$, unit circle is shown by solid curve (from [16])

Fig.6 Coarse-grained probability distribution for the eigenstates of the Google matrix for Linux Kernel version 2.6.32. The horizontal lines show the first 64 eigenvectors ordered vertically by $|\lambda _{i}|$ (from [16])

The Google matrix can be constructed also for other directed networks, e.g. for the procedure call network of the Linux Kernel software introduced in [15]. In this case the spectrum of $\lambda$ is described by the fractal Weyl law with the fractal dimension $d\approx 1.3$ (see Fig.5 from [16]). Numerical analysis shows that the eigenstates of matrix $G$ are localized (see Fig.6 from [16]). Arnoldi iteration method allows to compute many eigenvalues and eigenvectors for matrices of rather large size [13,14,16].

Other examples of $G$ matrix include the Google matrix of brain [17] and business process management [18], see also [19]. Applications of Google matrix analysis to DNA sequences is described in [20]. Such a Google matrix approach allows also to analyze entanglement of cultures via ranking of multilingual Wikipedia articles abouts persons [21]

Historical notes

The Google matrix with damping factor was described by Sergey Brin and Larry Page in 1998 [22], see also articles PageRank and [23],[24].

See also

• PageRank, CheiRank
• Arnoldi iteration
• Markov chain, Transfer operator, Perron–Frobenius theorem
• Web search engines

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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