Numeric examples of PageRank values in a small graph with a damping factor of 0.85. The exact solution is:

${\displaystyle \mathbf {R} ={\begin{bmatrix}PR(A)\\PR(B)\\\vdots \\PR(K)\end{bmatrix))={\dfrac {1}{579662461))\cdot {\begin{pmatrix}19002201\\222822800\\198772220\\22657320\\46886400\\22657320\\9372840\\9372840\\9372840\\9372840\\9372840\end{pmatrix))\approx {\begin{pmatrix}0,03278149\\0,38440095\\0,34291029\\0,03908709\\0,08088569\\0,03908709\\0,01616948\\0,01616948\\0,01616948\\0,01616948\\0,01616948\end{pmatrix))}$

And here’s the solution for any arbitrary damping factor d:

${\displaystyle \mathbf {R_{d)) ={\begin{bmatrix}PR_{d}(A)\\PR_{d}(B)\\\vdots \\PR_{d}(K)\end{bmatrix))=\left({\dfrac {1}{(d+1)\cdot (7\,d^{4}+28\,d^{2}+12\,d-132)))\right)\cdot {\begin{bmatrix}\left({d}+1\right)\left(7\,{d}^{3}+6\,{d}+12\right)\left({d}-1\right)\\-9\,{d}^{3}-18\,{d}^{2}-46\,{d}-12\\-11\,{d}^{4}-12\,{d}-12-18\,{d}^{3}-32\,{d}^{2}\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\12\,\left({d}+1\right)\left(4\,{d}^{2}-3\,{d}-1\right)\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\end{bmatrix))}$