How to rank pages on the web 
The model (III)
Let us denote \(r=\frac{1-p}{1-p\sum_{s_{j}\neq0}x_{j}}\). As \(p\neq1\), it follows that \[\begin{array}{ccl} 0 & \leq & \sum_{s_{j}=0}x_{j}\leq1\Rightarrow\\ & \Rightarrow & 1-p\leq1-p\sum_{s_{j}=0}x_{j}\leq1\Rightarrow\\ & \Rightarrow & \frac{1-p}{1}\leq\frac{1-p}{1-p\sum_{s_{j}\neq0}x_{j}}\leq\frac{1-p}{1-p}\Rightarrow\\ & \Rightarrow & 1-p\leq r\leq1. \end{array}\]
From the equation \[x_{i}=\frac{1-p\sum_{s_{j}\neq0}x_{j}}{n}+p\sum_{s_{j}\neq0}\frac{g_{ij}}{s_{j}}x_{j},\] by multiplying both side by \(rn\) (we have previously verified that \(r\neq0\)), we have that \[\begin{array}{ccl} rnx_{i} & = & rn\frac{1-p\sum_{s_{j}\neq0}x_{j}}{n}+rnp\sum_{s_{j}\neq0}\frac{g_{ij}}{s_{j}}x_{j},\\ rnx_{i} & = & \left(1-p\right)+p\sum_{s_{j}\neq0}\frac{g_{ij}}{s_{j}}rnx_{j}, \end{array}\] or, equivalently, \[rnx_{i}=\left(1-p\right)+p\left(\frac{rnx_{j1}}{s_{j1}}+\frac{rnx_{j2}}{s_{j12}}+\cdots+\frac{rnx_{jk}}{s_{jk}}\right),\] where \(j_{1},j_{2},\cdots,j_{k}\) are the indices of pages that include some link pointing towards page \(i\).
We thus derive that \(PR\left(P_{i}\right)=rnx_{i}\) and that the PageRank value is, up to multiplication by \(rn\), the long-run probability that the user is visiting page \(i\). Hence, the PageRank value of a web page is between \(0\) and \(rn\) (or, more specifically, between \(1-p\) and \(rn\)), and the sum of the PageRank values for all existing web pages is \(rn\) (thus, less than or equal to \(n\), being strictly less than \(n\) when \(r<1\), that is, when there exist web pages that do not include any link).
Note now that, if we now the PageRank values for every page, we can easily compute the values for the \(x_{i}.\)
Indeed, as \(PR\left(P_{i}\right)=rnx_{i}\), it holds that \[x_{i}=\frac{PR\left(P_{i}\right)}{rn}.\]
Moreover, as we have have verified that the sum of all the PageRank values is equal to \(rn\), if follows that \[x_{i}=\frac{PR\left(P_{i}\right)}{\sum_{i=1}^{n}PR\left(P_{i}\right)}\]
(that is, \(x_{i}\) is equal to the ratio bewteen the PageRank value of page \(i\) and the total PageRank value).
What about if we do not know the PageRank values for every existing web page?
