The damping factor is $d=\frac{4}{5}$ and the network is the following:
$1\rightarrow2,3\\2\rightarrow3,4\\3\rightarrow1,2,4,6\\4\\5\rightarrow2\\6\rightarrow5$
What I have done so far
The google matrix is defined as $L_d=d(L+\frac{1}{n}D)+(1-d)\frac{1}{n}S$.
$$d_{ij}= \left\{ \begin{array}{ll} 0 & deg_+(j)>0 \\ 1 & deg_+(j)=0 \\ \end{array} \right., s_{ij}=1$$ and $$l_{ij}= \left\{ \begin{array}{ll} \frac{1}{deg_+(j)} & i\in\Gamma_+(j) \\ 0 & i\notin\Gamma_+(j) \\ \end{array} \right.$$
From my calculations I got: $$ L_d= \left(\begin{matrix}0&0&\frac{7}{30}&\frac{1}{6}&0&0\\\frac{13}{30}&0&\frac{7}{30}&\frac{1}{6}&\frac{5}{6}&0\\\frac{13}{30}&\frac{13}{30}&0&\frac{1}{6}&0&0\\0&\frac{13}{30}&\frac{7}{30}&\frac{1}{6}&0&0\\0&0&0&\frac{1}{6}&0&\frac{5}{6}\\0&0&\frac{7}{30}&\frac{1}{6}&0&0\end{matrix}\right) $$
My questions
- Is the google matrix correctly calculated?
- How can I calculate the page rank? I found the definition "The page rank is the unique eigenvector of $L_d$ to the eigenvalue 1". However when I calculate the eigenvalues of the characteristic polynomial, there is no eigenvalue of 1. Then I tried to just use the eigenvalue 1 to calculate the vector, what resulted in the null vector. Where is my mistake?