91影视

Given that the mini-Web with the graph

$\begin{array}{ccc}1& 鈬�& 2\\ 鈫�& 鈫�& 鈫�\\ 3& 鈫�& 4\end{array}$

Consider the surfing model with a 鈥渏umping rate鈥� of 20%, as discussed in Exercise 2.1.53. The corresponding transition matrix is

.$B=\left[\begin{array}{cccc}0.05& 0.45& 0.05& 0.05\\ 0.45& 0.05& 0.05& 0.85\\ 0.45& 0.45& 0.05& 0.05\\ 0.05& 0.05& 0.85& 0.05\end{array}\right]$

This transition matrix is positive and therefore regular, so that Theorem 2.3.11 applies.

The transition matrix is as follows:

$B=\left[\begin{array}{cccc}0.05& 0.45& 0.05& 0.05\\ 0.45& 0.05& 0.05& 0.85\\ 0.45& 0.45& 0.05& 0.05\\ 0.05& 0.05& 0.85& 0.05\end{array}\right]$

Find equilibrium distribution using MATLAB.

Enter the command:

>>B=[0.05, 0.45, 0.05, 0.05;

0.45, 0.05, 0.05, 0.85;

0.45, 0.45, 0.05, 0.05;

0.05, 0.05, 0.85, 0.05];

>>format rat

>>B^30

Which yields

$\begin{array}{l}\frac{5}{28},\text{鈥�}\frac{5}{28},\text{鈥�}\frac{5}{28},\text{鈥�}\frac{5}{28}\\ \frac{9}{28},\text{鈥�}\frac{9}{28},\text{鈥�}\frac{9}{28},\text{鈥�}\frac{9}{28}\\ \frac{1}{4},\text{鈥�}\frac{1}{4},\text{鈥�}\frac{1}{4},\text{鈥�}\frac{1}{4}\\ \frac{1}{4},\text{鈥�}\frac{1}{4},\text{鈥�}\frac{1}{4},\text{鈥�}\frac{1}{4}\end{array}$

Thus the equilibrium distribution is:

${\stackrel{鈫�}{x}}_{equ}=\left[\begin{array}{l}\frac{5}{28}\\ \frac{9}{28}\\ \frac{1}{4}\\ \frac{1}{4}\end{array}\right]$