# Before we begin, let's load the libraries.
%pylab notebook
import numpy as np
import numpy.linalg as la
from readonly.PageRankFunctions import *
np.set_printoptions(suppress=True)

```
Populating the interactive namespace from numpy and matplotlib
```

# Replace the ??? here with the probability of clicking a link to each website when leaving Website F (FaceSpace).
L = np.array([[0, 1/2, 1/3, 0, 0, 0, ],
[1/3, 0, 0, 0, 1/2, 0, ],
[1/3, 1/2, 0, 1, 0, 1/2, ],
[1/3, 0, 1/3, 0, 1/2, 1/2, ],
[0, 0, 0, 0, 0, 0, ],
[0, 0, 1/3, 0, 0, 0, ]])

eVals, eVecs = la.eig(L) # Gets the eigenvalues and vectors
order = np.absolute(eVals).argsort()[::-1] # Orders them by their eigenvalues
eVals = eVals[order]
eVecs = eVecs[:,order]
r = eVecs[:, 0] # Sets r to be the principal eigenvector
100 * np.real(r / np.sum(r)) # Make this eigenvector sum to one, then multiply by 100 Procrastinating Pats

r = 100 * np.ones(6) / 6 # Sets up this vector (6 entries of 1/6 × 100 each)
r # Shows it's value

r = L @ r # Apply matrix L to r
r # Show it's value
# Re-run this cell multiple times to converge to the correct answer.

r = 100 * np.ones(6) / 6 # Sets up this vector (6 entries of 1/6 × 100 each)
for i in np.arange(100) : # Repeat 100 times
r = L @ r
r

r = 100 * np.ones(6) / 6 # Sets up this vector (6 entries of 1/6 × 100 each)
lastR = r
r = L @ r
i = 0
while la.norm(lastR - r) > 0.01 :
lastR = r
r = L @ r
i += 1
print(str(i) + " iterations to convergence.")
r

```
18 iterations to convergence.
```

# We'll call this one L2, to distinguish it from the previous L.
L2 = np.array([[0, 1/2, 1/3, 0, 0, 0, 0 ],
[1/3, 0, 0, 0, 1/2, 0, 0 ],
[1/3, 1/2, 0, 1, 0, 1/3, 0 ],
[1/3, 0, 1/3, 0, 1/2, 1/3, 0 ],
[0, 0, 0, 0, 0, 0, 0 ],
[0, 0, 1/3, 0, 0, 0, 0 ],
[0, 0, 0, 0, 0, 1/3, 1 ]])

r = 100 * np.ones(7) / 7 # Sets up this vector (6 entries of 1/6 × 100 each)
lastR = r
r = L2 @ r
i = 0
while la.norm(lastR - r) > 0.01 :
lastR = r
r = L2 @ r
i += 1
print(str(i) + " iterations to convergence.")
r

```
131 iterations to convergence.
```

d = 0.5 # Feel free to play with this parameter after running the code once.
M = d * L2 + (1-d)/7 * np.ones([7, 7]) # np.ones() is the J matrix, with ones for each entry.

r = 100 * np.ones(7) / 7 # Sets up this vector (6 entries of 1/6 × 100 each)
lastR = r
r = M @ r
i = 0
while la.norm(lastR - r) > 0.01 :
lastR = r
r = M @ r
i += 1
print(str(i) + " iterations to convergence.")
r

```
8 iterations to convergence.
```

# PACKAGE
# Here are the imports again, just in case you need them.
# There is no need to edit or submit this cell.
import numpy as np
import numpy.linalg as la
from readonly.PageRankFunctions import *
np.set_printoptions(suppress=True)

# GRADED FUNCTION
# Complete this function to provide the PageRank for an arbitrarily sized internet.
# I.e. the principal eigenvector of the damped system, using the power iteration method.
# (Normalisation doesn't matter here)
# The functions inputs are the linkMatrix, and d the damping parameter - as defined in this worksheet.
# (The damping parameter, d, will be set by the function - no need to set this yourself.)
def pageRank(linkMatrix, d) :
n = linkMatrix.shape[0]
M = d * linkMatrix + (1-d)/n * np.ones([n, n])
r = 100 * np.ones(n) / n # Sets up this vector (6 entries of 1/6 × 100 each)
lastR = r
r = M @ r
i = 0
while la.norm(lastR - r) > 0.01 :
lastR = r
r = M @ r
i += 1
print(str(i) + " iterations to convergence.")
return r

# Use the following function to generate internets of different sizes.
generate_internet(5)

# Test your PageRank method against the built in "eig" method.
# You should see yours is a lot faster for large internets
L = generate_internet(10)

pageRank(L, 1)

```
11 iterations to convergence.
```

# Do note, this is calculating the eigenvalues of the link matrix, L,
# without any damping. It may give different results that your pageRank function.
# If you wish, you could modify this cell to include damping.
# (There is no credit for this though)
eVals, eVecs = la.eig(L) # Gets the eigenvalues and vectors
order = np.absolute(eVals).argsort()[::-1] # Orders them by their eigenvalues
eVals = eVals[order]
eVecs = eVecs[:,order]
r = eVecs[:, 0]
100 * np.real(r / np.sum(r))

# You may wish to view the PageRank graphically.
# This code will draw a bar chart, for each (numbered) website on the generated internet,
# The height of each bar will be the score in the PageRank.
# Run this code to see the PageRank for each internet you generate.
# Hopefully you should see what you might expect
# - there are a few clusters of important websites, but most on the internet are rubbish!
%pylab notebook
r = pageRank(generate_internet(100), 0.9)
plt.bar(arange(r.shape[0]), r);

```
Populating the interactive namespace from numpy and matplotlib
35 iterations to convergence.
```