Page rank via eigenvectors :

A={{0,0,1/2,1},
{1/3,0,0,0},
{1/3,1/2,0,0},
{1/3,1/2,1/2,0}};
Eigenvalues[A]

Eigensystem[A]

eigenvector x for eigenvalue 1 (vector of ranks for sites 1,2,3,4) normalized to add up to 1 (100%)

x={12,4,6,9}/31 //N

Page rank via dynamical system

Start with equally distributed ranks v for all 4 sites.

v={1/4,1/4,1/4,1/4};
A.v//N

After 1 step,now 37.5% of the visitors are at site 1.

A.A.v//N

MatrixPower[A,4].v//N

MatrixPower[A,8].v//N

After 8 steps the distribution of visitors is equal to the eigenvector computed above in 3 digits.