We will consider a simple model of the Internet that has three pages and links between them as shown here. For instance, page 1 links to both pages 2 and 3, but page 2 only links to page 1.

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Figure4.5.7Our first Internet.

We will measure the quality of the \(j^{th}\) page with a number \(x_j\text{,}\) which is called the PageRank of page \(j\text{.}\) The PageRank is determined by the following rule: each page divides its PageRank into equal pieces, one for each outgoing link, and gives one piece to each of the pages it links to. A page's PageRank is the sum of all the PageRank it receives from pages linking to it.

For instance, page 3 has two outgoing links. It therefore divides its PageRank \(x_3\) in half and gives half to page 1. Page 2 has only one outgoing link so it gives all of its PageRank \(x_2\) to page 1. We therefore have

\begin{equation*} x_1 = x_2 + \frac12 x_3 \text{.} \end{equation*}

  1. Find similar expressions for \(x_2\) and \(x_3\text{.}\)

  2. We now form the PageRank vector \(\xvec = \threevec{x_1}{x_2}{x_3}\text{.}\) Find a matrix \(G\) such that the expressions for \(x_1\text{,}\) \(x_2\text{,}\) and \(x_3\) can be written in the form \(G\xvec = \xvec\text{.}\) The matrix \(G\) is called the “Google matrix”.

  3. Explain why \(G\) is a stochastic matrix.

  4. Since \(\xvec\) is defined by the equation \(G\xvec = \xvec\text{,}\) any vector in the eigenspace \(E_1\) satisfies this equation. So that we might work with a specific vector, we will define the PageRank vector to be the steady-state vector of the stochastic matrix \(G\text{.}\) Find this steady state vector.

  5. The PageRank vector \(\xvec\) is composed of the PageRanks for each of the three pages. Which page of the three is assessed to have the highest quality? By referring to the structure of this small model of the Internet, explain why this is a good choice.

  6. If we begin with the initial vector \(\xvec_0 = \threevec{1}{0}{0}\) and form the Markov chain \(\xvec_{k+1} = G\xvec_k\text{,}\) what does the Perron-Frobenius theorem tell us about the long-term behavior of the Markov chain?

  7. Verify that this Markov chain converges to the steady-state PageRank vector.