I would like to understand the matrix equation $(4)$ on the page $95$ here concerning some Google matrix computations. I got this far:
$$||x(t)||=||dWx(t-1)||+\frac{||x(t-1)||-||dWx(t-1)||}{N}\cdot N=||x(t-1)||\ {\stackrel{\mbox{?}}{=}}\ 1,$$ but why this last is $= 1$ ?
Just putting up my comments above as an answer.
Basically, equation $4$ specifies an iterative scheme to find the fixed point of the PageRank equation specified by Brin et al., and it works along the lines of the Banach fixed point theorem.
That is, given the equation (on vectors) $\mathbf x = d\mathbf{Wx} + \frac{\|\mathbf x\|_1 - \|d\mathbf{Wx}\|_1}{N}\mathbf{1}_N$ , we take our function $f(\mathbf x) = d\mathbf{Wx} + \frac{\|\mathbf x\|_1 - \|d\mathbf{Wx}\|_1}{N}\mathbf{1}_N$ and form a sequence like so : take any starting vector $\mathbf x(0)$ so that $\|\mathbf x(0)\|_1 = 1$. Now define $\mathbf x(t) = f(\mathbf x(t-1))$ : so you keep applying $f$ to what you get. The calculations you make show by induction that $\|\mathbf x(t)\|_{1} = 1$ for all $t$.
This gives a sequence of vectors $\{\mathbf x(t)\}_{t=0,1,2,...}$, and the point is, under conditions given by the Banach FPT, this converges to a solution vector $\mathbf x$, which will satisfy $\mathbf x = f(\mathbf x)$, the PageRank equation.
Why is that? Well, you kept applying $f$ again and again : suppose that $f$ was a contraction i.e. it tended to bring two points closer upon applying $f$ on each of them. Then, repeated application of $f$ keeps bringing the functions values closer to each other, and finally we come to a point where applying $f$ doesn't change the point (which is a fixed point). The condition $d <1$ ensures that Banach FPT applies, giving a solution. Thus, existence and uniqueness of solution is given by the Banach FPT.
Furthermore, by closure, that solution will also satisfy $\|\mathbf x\|_1 = 1$, which is desirable to Brin, I would think.