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

${\displaystyle 𝐑=\left[\begin{array}{c}PR(A)\\ PR(B)\\ ⋮\\ PR(K)\end{array}\right]={\displaystyle \frac{1}{579662461}}\cdot \left(\begin{array}{c}19002201\\ 222822800\\ 198772220\\ 22657320\\ 46886400\\ 22657320\\ 9372840\\ 9372840\\ 9372840\\ 9372840\\ 9372840\end{array}\right)\approx \left(\begin{array}{c}0,03278149\\ 0,38440095\\ 0,34291029\\ 0,03908709\\ 0,08088569\\ 0,03908709\\ 0,01616948\\ 0,01616948\\ 0,01616948\\ 0,01616948\\ 0,01616948\end{array}\right)}$

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

${\displaystyle {𝐑}_{𝐝}=\left[\begin{array}{c}P{R}_d(A)\\ P{R}_d(B)\\ ⋮\\ P{R}_d(K)\end{array}\right]=\left({\displaystyle \frac{1}{(d+1)\cdot (7d^4+28d^2+12d-132)}}\right)\cdot \left[\begin{array}{c}(d+1)(7d^3+6d+12)(d-1)\\ -9d^3-18d^2-46d-12\\ -11d^4-12d-12-18d^3-32d^2\\ 2(d+1)(7d^2+2d+6)(d-1)\\ 12(d+1)(4d^2-3d-1)\\ 2(d+1)(7d^2+2d+6)(d-1)\\ -2(d+1)(-6+d^2)(d-1)\\ -2(d+1)(-6+d^2)(d-1)\\ -2(d+1)(-6+d^2)(d-1)\\ -2(d+1)(-6+d^2)(d-1)\\ -2(d+1)(-6+d^2)(d-1)\end{array}\right]}$