Question

7. Bonus [3+3+4pts] Before answering this question, read the Google page rank article on Pi- azza in the 'General Resources' section. The Google page rank algorithm has a lot to do with the eigenvector corresponding to the largest eigenvalue of a so-called stochastic matrix, which describes the links between websites.2 Stochastic matrices have non-negative entries and each column sums to1, and one can show (under a few technical assumptions) that it has the eigen- values λι-1 > A1 2 lhol. Thus, we can use the power method3 to find the eigenvector u corresponding to λ, which can be shown to have either all negative or all positive entries. These entries can be interpreted as the importance of individual websites. Let us construct a large stochastic matrices (pick a size n 2 100, the size of our "toy internet") in MATLAB as follows: Note that for MATLAB functions that do not use external libraries, you can see how they are implemented by typing edit name of function. Doing that for the roots function will show you that MATLAB implements root finding exactly using a companion matrix as described above See the interesting 2006 SIREV paper The 25,000,000,000 eigenvector. Kurt Bryan and Tanya Leise. It's easy to find-just google it! The linear algebra behind Google by We have discussed the power method for symmetric matrices, but it also works for non-symmetric matrices. Ieye(n) A-0.5*I(randperm(n),:) + (max(2,randn(n,n))-2); A -A - diag(diag(A)); L-Adiag(1./(max(1e-10,sum(A,1)))); (a) Plot the sparsity structure of L (i.e., the nonzero entries in the matrix) using the command spy. Each non-zero entry corresponds to a link between two websites. (b) Plot the (complex) eigenvalues of I by plotting the real part of the eigenvalues on the r-axis, and the imaginary part on the y-axis.4 Additionally, plot the unit circle and check that all eigenvalues are inside the unit circle, but λί (c) The matrix L contains many zeros. One of the technical assumptions for proving theorems is that all entries in L are positive. As a remedy, one considers the matrices S KL+ (1-K)E where E is a matrix with entries 1/n in every component. Study the influence of k numerically by visualizing the eigenvalues of S for different values of K. Why will K< 1 improve the speed of convergence of the power method? Please also hand in your code.

Can you help me with this question please? For the code, please do it on MATLAB. Thanks

Answer 1

clc
clear
close all

n = 100; % any integer >=100

I = eye(n);
A = 0.5*I(randperm(n),:)+(max(2,randn(n,n))-2);
A = A - diag(diag(A));
L = A*diag(1./(max(1e-10,sum(A,1))));

spy(L)
title('sparsity structure of L')

% computing eigen values of L
eigValuesOfL = eig(L)
realPartEigValuesOfL = real(eigValuesOfL);
imagPartEigValuesOfL = imag(eigValuesOfL);

%plotting imag vs real
figure
plot(realPartEigValuesOfL,imagPartEigValuesOfL,'*')

%computing and plotting the unit circle centered at (0,0)
x = linspace(-1,1,100);
yUpper = sqrt(1-x.^2); % x^2+y^2=1
yLower = -yUpper;
hold on
plot(x,yUpper,'-r')
hold on
plot(x,yLower,'-r')
axis equal
xlabel('real part of eigne values of L')
ylabel('immaginary part of eigne values of L')
title('Eigen values of L')

% matrix E of all values = 1/n
E = ones(n,n).*1/n;

% let k start at 0.1 increase by 0.1 upto 3
K = 0.1:0.1:3;
for i=1:numel(K)
k = K(i);
S = k*L + (1-k)*E;
eigValuesOfS = eig(S);

% we can determine the rate of convergence by comparing the ratio of the
% first to second largest eigen value for each value k

% for purposes of computing the cpnvergence rate
sizeOfEachEignValue = abs(eigValuesOfS);
sortedEig = sort(sizeOfEachEignValue);

% ratio of largest to second largest eignen value
ratio(i) = sortedEig(end)/sortedEig(end-1);
end

figure
plot(K,ratio,'linewidth',2)
xlabel('k')
ylabel('rate of convergence')
grid on

% the rate of convergence is seen to exponentially decrease as k tends to 1
% and remains almost constant and low after k=1 so values of k<1 will increase the
% speed of convergence of the power method

sparsity structure of L 10 20 50 60 70 80 100 20 40 60 80 100 nz = 304

Eigen values of L 0.8 0.6 0.4 0.2 米米米米 米米 米米 米米 兼米 米米 米米米米 米米米米 米米 0.2 2 米 米 米米 米米 米米米米 0.6 -0.8 0.5 0.5 real part of eigne values of L

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