Commit 783c7ed1 by Jim Hefferon

### another page rank exercise

parent 4047c798
 ... ... @@ -137,6 +137,11 @@ Web Search Engine}, \url{http://infolab.stanford.edu/pub/papers/google.pdf}, retrieved Feb.~2012. \bibitem[Bryan \& Leise]{BryanLeise} Kurt Bryan, Tanya Leise, \emph{The \$25,000,000,000 Eigenvector: the Linear Algebra Behind Google}, SIAM Review, Vol.\ 48, no.\ 3 (2006), p.\ 569-81. \bibitem[Casey]{Casey} John Casey, \emph{The Elements of Euclid, Books~I to~VI and~XI}, ... ... This diff is collapsed.  ... ... @@ -29479,6 +29479,66 @@ ans = 0.017398 =\alpha\cdot 1+(1-\alpha)\cdot 1$, which is one. \end{ans} \begin{ans}{2} We have this. \begin{equation*} H=\begin{mat} 0 &0 &1 &1/2 \\ 1/3 &0 &0 &0 \\ 1/3 &1/2 &0 &1/2 \\ 1/3 &1/2 &0 &0 \end{mat} \end{equation*} \begin{exparts} \item This \textit{Sage} session gives the answer. \begin{lstlisting} sage: H=matrix(QQ,[[0,0,1,1/2], [1/3,0,0,0], [1/3,1/2,0,1/2], [1/3,1/2,0,0]]) sage: S=matrix(QQ,[[1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4]]) sage: I=matrix(QQ,[[1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]]) sage: alpha=0.85 sage: G=alpha*H+(1-alpha)*S sage: N=G-I sage: 1200*N [-1155.00000000000 45.0000000000000 1065.00000000000 555.000000000000] [ 385.000000000000 -1155.00000000000 45.0000000000000 45.0000000000000] [ 385.000000000000 555.000000000000 -1155.00000000000 555.000000000000] [ 385.000000000000 555.000000000000 45.0000000000000 -1155.00000000000] sage: M=matrix(QQ,[[-1155,45,1065,555], [385,-1155,45,45], [385,555,-1155,555], [385,555,45,-1155]]) sage: M.echelon_form() [ 1 0 0 -106613/58520] [ 0 1 0 -40/57] [ 0 0 1 -57/40] [ 0 0 0 0] sage: v=vector([106613/58520,40/57,57/40,1]) sage: (v/v.norm()).n() (0.696483066294572, 0.268280959381099, 0.544778023143244, 0.382300367118066) \end{lstlisting} \item Continue the \textit{Sage} to get this. \begin{lstlisting} sage: alpha=0.95 sage: G=alpha*H+(1-alpha)*S sage: N=G-I sage: 1200*N [-1185.00000000000 15.0000000000000 1155.00000000000 585.000000000000] [ 395.000000000000 -1185.00000000000 15.0000000000000 15.0000000000000] [ 395.000000000000 585.000000000000 -1185.00000000000 585.000000000000] [ 395.000000000000 585.000000000000 15.0000000000000 -1185.00000000000] sage: M=matrix(QQ,[[-1185,15,1155,585], [395,-1185,15,15], [395,585,-1185,585], [395,585,15,-1185]]) sage: M.echelon_form() [ 1 0 0 -361677/186440] [ 0 1 0 -40/59] [ 0 0 1 -59/40] [ 0 0 0 0] sage: v=vector([361677/186440,40/59,59/40,1]) sage: (v/v.norm()).n() (0.713196892748114, 0.249250262646952, 0.542275102671275, 0.367644137404254) \end{lstlisting} \item Page $p_3$ is important, but it passes its importance on to only one page, $p_1$. So that page receives a large boost. \end{exparts} \end{ans} \subsection{Topic: Linear Recurrences} \begin{ans}{1}
 ... ... @@ -373,22 +373,22 @@ beginfig(9); % m %numeric u; %scaling factor %numeric v; %vertical scaling factor %numeric w; %horizontal scaling factor numeric circlescale; circlescale=20pt; numeric circlescale; circlescale=19pt; path node; node=fullcircle scaled circlescale; path n[]; % node paths pickup pencircle scaled line_width_light; z1=(0w,6v); n1=node shifted z1; draw n1; label(btex $p_1$ etex,z1); draw n1; label(btex \small $p_1$ etex,z1); z2=(9w,y1); n2=node shifted z2; draw n2; label(btex $p_2$ etex,z2); draw n2; label(btex \small $p_2$ etex,z2); z3=(x2,0v); n3=node shifted z3; draw n3; label(btex $p_3$ etex,z3); draw n3; label(btex \small $p_3$ etex,z3); z4=(x1,y3); n4=node shifted z4; draw n4; label(btex $p_4$ etex,z4); draw n4; label(btex \small $p_4$ etex,z4); path p[], q[]; pair times[]; % intersection times % arrow from p1 to p2 ... ... @@ -414,6 +414,63 @@ beginfig(9); % m endfig; % Exercise from KURT BRYAN AND TANYA LEISE % beginfig(10); % m %numeric u; %scaling factor %numeric v; %vertical scaling factor %numeric w; %horizontal scaling factor numeric circlescale; circlescale=19pt; path node; node=fullcircle scaled circlescale; path n[]; % node paths pickup pencircle scaled line_width_light; z1=(0w,6v); n1=node shifted z1; draw n1; label(btex \small $p_1$ etex,z1); z2=(9w,y1); n2=node shifted z2; draw n2; label(btex \small $p_2$ etex,z2); z3=(x2,0v); n3=node shifted z3; draw n3; label(btex \small $p_3$ etex,z3); z4=(x1,y3); n4=node shifted z4; draw n4; label(btex \small $p_4$ etex,z4); path p[], q[]; pair times[]; % intersection times % arrow from p1 to p2 p12=z1--z2; times121=p12 intersectiontimes n1; times122=p12 intersectiontimes n2; drawarrow subpath(xpart(times121)+.05,xpart(times122)-.05) of p12; % arrow from p1 to p3 p13=z1--z3; times131=p13 intersectiontimes n1; times133=p13 intersectiontimes n3; drawdblarrow subpath(xpart(times131)+.05,xpart(times133)-.05) of p13; % arrow from p1 to p4 p14=z1--z4; times141=p14 intersectiontimes n1; times144=p14 intersectiontimes n4; drawdblarrow subpath(xpart(times141)+.05,xpart(times144)-.05) of p14; % arrow from p2 to p3 p23=z2--z3; times232=p23 intersectiontimes n2; times233=p23 intersectiontimes n3; drawarrow subpath(xpart(times232)+.05,xpart(times233)-.05) of p23; % arrow from p2 to p4 p24=z2--z4; times242=p24 intersectiontimes n2; times244=p24 intersectiontimes n4; drawarrow subpath(xpart(times242)+.05,xpart(times244)-.05) of p24; % arrow from p4 to p5 p43=z4--z3; times433=p43 intersectiontimes n4; times434=p43 intersectiontimes n3; drawarrow subpath(xpart(times433)+.05,xpart(times434)-.05) of p43; endfig; end \ No newline at end of file
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 ... ... @@ -35,7 +35,7 @@ Today's statistics would be quite different, both because of inflation and because of technical changes in the industries.) \begin{center} \begin{tabular}{r|rrrr} \multicolumn{1}{c|}{\ } %put the | in the right place \multicolumn{1}{c}{\ } %put the | in the right place &\multicolumn{1}{r}{\begin{tabular}[b]{@{}c@{}} \textit{used by} \\[-.5ex] \textit{steel} \end{tabular}} ... ... @@ -46,7 +46,7 @@ inflation and because of technical changes in the industries.) \textit{used by} \\[-.5ex] \textit{others} \end{tabular}} &\textit{total} \\ \hline \cline{2-5} \begin{tabular}{r} \textit{value of} \\[-.5ex] \textit{steel} \end{tabular} &$5\,395$ &$2\,664$ & &$25\,448$ \\ \begin{tabular}{r} \textit{value of} \\[-.5ex] \textit{auto} \end{tabular} ... ...
 ... ... @@ -160,6 +160,7 @@ $\alpha$'s. % Numbers computed as: % sage: H=matrix(QQ,[[0,0,1/3,1/4], [1,0,1/3,1/4], [0,1,0,1/4], [0,0,1/3,1/4]]) % sage: S=matrix(QQ,[[1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4]]) % sage: I=matrix(QQ,[[1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]]) % sage: alpha=0.95 % sage: G=alpha*H+(1-alpha)*S % sage: N=G-I ... ... @@ -203,6 +204,78 @@ Two additional excellent expositions are =\alpha\cdot 1+(1-\alpha)\cdot 1$, which is one. \end{answer} \item \cite{BryanLeise} Give the importance ranking for this web of pages. \begin{center} \includegraphics{ch5.10} \end{center} \begin{exparts} \item Use$\alpha=0.85$. \item Use$\alpha=0.95$. \item Observe that while$p_3$is linked-to from all other pages, and therefore seems important, it is not the highest ranked page. What is the highest ranked page? Explain. \end{exparts} \begin{answer} We have this. \begin{equation*} H=\begin{mat} 0 &0 &1 &1/2 \\ 1/3 &0 &0 &0 \\ 1/3 &1/2 &0 &1/2 \\ 1/3 &1/2 &0 &0 \end{mat} \end{equation*} \begin{exparts} \item This \textit{Sage} session gives the answer. \begin{lstlisting} sage: H=matrix(QQ,[[0,0,1,1/2], [1/3,0,0,0], [1/3,1/2,0,1/2], [1/3,1/2,0,0]]) sage: S=matrix(QQ,[[1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4], [1/4,1/4,1/4,1/4]]) sage: I=matrix(QQ,[[1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]]) sage: alpha=0.85 sage: G=alpha*H+(1-alpha)*S sage: N=G-I sage: 1200*N [-1155.00000000000 45.0000000000000 1065.00000000000 555.000000000000] [ 385.000000000000 -1155.00000000000 45.0000000000000 45.0000000000000] [ 385.000000000000 555.000000000000 -1155.00000000000 555.000000000000] [ 385.000000000000 555.000000000000 45.0000000000000 -1155.00000000000] sage: M=matrix(QQ,[[-1155,45,1065,555], [385,-1155,45,45], [385,555,-1155,555], [385,555,45,-1155]]) sage: M.echelon_form() [ 1 0 0 -106613/58520] [ 0 1 0 -40/57] [ 0 0 1 -57/40] [ 0 0 0 0] sage: v=vector([106613/58520,40/57,57/40,1]) sage: (v/v.norm()).n() (0.696483066294572, 0.268280959381099, 0.544778023143244, 0.382300367118066) \end{lstlisting} \item Continue the \textit{Sage} to get this. \begin{lstlisting} sage: alpha=0.95 sage: G=alpha*H+(1-alpha)*S sage: N=G-I sage: 1200*N [-1185.00000000000 15.0000000000000 1155.00000000000 585.000000000000] [ 395.000000000000 -1185.00000000000 15.0000000000000 15.0000000000000] [ 395.000000000000 585.000000000000 -1185.00000000000 585.000000000000] [ 395.000000000000 585.000000000000 15.0000000000000 -1185.00000000000] sage: M=matrix(QQ,[[-1185,15,1155,585], [395,-1185,15,15], [395,585,-1185,585], [395,585,15,-1185]]) sage: M.echelon_form() [ 1 0 0 -361677/186440] [ 0 1 0 -40/59] [ 0 0 1 -59/40] [ 0 0 0 0] sage: v=vector([361677/186440,40/59,59/40,1]) sage: (v/v.norm()).n() (0.713196892748114, 0.249250262646952, 0.542275102671275, 0.367644137404254) \end{lstlisting} \item Page$p_3$is important, but it passes its importance on to only one page,$p_1\$. So that page receives a large boost. \end{exparts} \end{answer} \end{exercises} \index{page ranking|)} \endinput ... ...
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