(Solved): only q4 and q5 Exercise IV: How to be famous with math How is the popularity o ...

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Exercise IV: How to be famous with math How is the popularity of a WebPage (or of Instagram accounts, people, scientific works, etc) calculated? A simple solution is to count how many times it is referenced by other pages. However, this is easy to cheat: one could just create millions of pages that reference a given page. A better idea is to say that a page is popular if it is referenced by popular pages. You may notice that this is a circular definition! However, there are ways to make this precise. For instance, the famous algorithm PageRank that made Google founders Sergey Brin and Larry Page famous (and fabulously rich) uses this idea. In this exercise consider a very simple model with three pages $P_1,P_2,P_3$, where - $P_1$ references $P_2$ only; - $P_2$ references $P_1$ and $P_3$; - $P_3$ references $P_1$ and $P_2$. The popularity of page $P_1,P_2,P_3$ is denoted by $x,y,z$. 1. (Preliminary question) If $A$ is $n\times n$ and invertible, what is the only solution to the system $AX=0_{n\times 1}$ ? 2. As a first model, assume that the popularity of each page is the sum of the popularity of each page that it is referenced by. This gives rise to the equations $?\begin{array}{l}x=y+z\\ y=x+z\\ z=y\end{array}$ Find the solution to this system in any way that you want. 3. You should see that the previous model is not very satisfying! Instead, assume that the popularity of each page is a fraction $?$ of the sum of the popularity of each page that it is referenced by. This gives rise to the equations $?\begin{array}{l}x=?(y+z)\\ y=?(x+z)\\ z=?y.\end{array}$ Find the solution to this system in any way that you want for $?=0.1$ (each page gets $10\%$ of the popularity of the pages it is referenced by). 4. Under what condition on $?$ can the system have a nontrivial solution (that is a solution other than $x=y=z=0$ )? You are asked to give an equation involving $?$, but there is no need to solve it. 5. You could use a calculator to estimate the values of $?$ in the previous question and then solve the system. If you do (try it!), you will see that the calculator still only gives you the trivial solution $x=y=z=0$. Why? There are other ways to find nontrivial solutions, but this is outside of the scope of this class.