It doesn't require linear algebra to understand the paper or how the algorithm works, but it does require linear algebra to understand why the algorithm works. In general, since the induced 1-norm of a stochastic matrix S is exactly equal to 1 but not smaller than 1, the mapping x↦Sx is NOT a contraction. Neither convergence of the power method nor uniqueness of fixed point are guaranteed. (If there are multiple fixed points, there are multiple inconsistent rankings.)

In the paper, the significance of the so-called "damping factor" is not clear. However, with linear algebra, we know that the damping factor makes the stochastic matrix positive rather than merely nonnegative. Hence the Perron eigenvalue is "simple" (i.e. of multiplicity one), the Perron vector is unique and the power iteration must converge to it.