Column stochastic matrix/Positive row/Network/Vertex hierarchy/Remark
In the situation of fact, we can find the eigendistribution by solving a system of linear equations. If we are dealing with a huge matrix (think about vertices), then such a computation time-consuming. Often, it is not necessary to know the eigendistribution precisely; it is enough to know a good approximation. For this, we may start with an arbitrary distribution, and we compute finitely many iterations. Because of fact, we know that this method gives arbitrarily good approximations of the eigendistribution. For example, a search engine for the web generates for search item an ordered list of web pages where this search item occurs. How does this ordering arise? The true answer is, at least for the first entries, that it depends on how much someone has paid. Despite of this, it is a natural approach, and this is also the basis for the Page ranks, to consider the numerical ordering in the eigendistribution. The first entry is the one where most people would "finally“ end up when they follow with the same probability any possible link. This movement is modelled by the stochastic matrix described in example.
