We model the web by a directed graph with $n$ vertices. The matrix $A$ is the stochastic matrix described in question 29. We define $$B = ( 1 - \alpha ) A + \frac { \alpha } { n } J _ { n }$$ where $J _ { n }$ is the matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ whose coefficients are all equal to $1$, $A$ is the stochastic matrix described in question 29 and $\alpha$ is a real number in $] 0,1 [$, called the damping factor. Let $Q$ be a probability distribution. We define the sequence $\left( Q ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ by $Q ^ { ( k ) } = Q B ^ { k }$ for every natural number $k$. Prove that the sequence $\left( Q ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges and that its limit $Q ^ { \infty }$ satisfies conditions (i) and (ii) described in the introduction to this part. It thus provides relevance scores for the $n$ pages of the web. The relevance of each page $j$ will be expressed as a function of those of the pages pointing to it, distinguishing between pages that point to another page and the others.
We model the web by a directed graph with $n$ vertices. The matrix $A$ is the stochastic matrix described in question 29. We define
$$B = ( 1 - \alpha ) A + \frac { \alpha } { n } J _ { n }$$
where $J _ { n }$ is the matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ whose coefficients are all equal to $1$, $A$ is the stochastic matrix described in question 29 and $\alpha$ is a real number in $] 0,1 [$, called the damping factor.
Let $Q$ be a probability distribution. We define the sequence $\left( Q ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ by $Q ^ { ( k ) } = Q B ^ { k }$ for every natural number $k$. Prove that the sequence $\left( Q ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges and that its limit $Q ^ { \infty }$ satisfies conditions (i) and (ii) described in the introduction to this part. It thus provides relevance scores for the $n$ pages of the web. The relevance of each page $j$ will be expressed as a function of those of the pages pointing to it, distinguishing between pages that point to another page and the others.