This formalizes the intuition in the previous section. Note that the rank of a page is divided
among its forward links evenly to contribute to the ranks of the pages they p\cite{Brin_Page}oint to. Note that \(c<1\) because there are a number of pages with no forward links and their weight is lost from the
system (see section 2.7). The equation is recursive but it may be computed by starting with any set
of ranks and iterating the computation until it converges. Figure 2 demonstrates the propagation
of rank from one pair of pages to another. Figure 3 shows a consistent steady state solution for a
set of pages.
Stated another way, let \(A\) be a square matrix with the rows and column corresponding to web
pages. Let \(A_{u,v}=\frac{1}{N_u}\) if there is an edge from \(u\) to \(v\) and \(A_{u,v}=0\) if not. If we treat \(R\) as a
vector over web pages, then we have \(R=cAR\). So \(R\) is an eigenvector of \(A\) with eigenvalue \(c\). In
fact, we want the dominant eigenvector of \(A\). It may be computed by repeatedly applying \(A\) to any
nondegenerate start vector.
There is a small problem with this simplified ranking function. Consider two web pages that
point to each other but to no other page. And suppose there is some web page which points to
one of them. Then, during iteration, this loop will accumulate rank but never distribute any rank
(since there are no outedges). The loop forms a sort of trap which we call a rank sink.
To overcome this problem of rank sinks, we introduce a rank source:
Definition 1 Let \(E\left(u\right)\) be some vector over the Web pages that corresponds to a source of rank.
Then, the PageRank of a set of Web pages is an assignment, \(R'\), to the Web pages which satisfies
\(R'\left(u\right)=c\sum_{v\in B_u}^{ }\frac{R'\left(v\right)}{N_v}+cE\left(u\right)\)
such that \(c\) is maximized and \(\left||R'\right||_1=1\ \left(\left|R'\right|\right|_1\) denotes the \(L_1\) norm of \(R'\)).