# --------------
# Code starts here

import numpy as np

# Code starts here

# Adjacency matrix
adj_mat = np.array([[0,0,0,0,0,0,1/3,0],
                   [1/2,0,1/2,1/3,0,0,0,0],
                   [1/2,0,0,0,0,0,0,0],
                   [0,1,0,0,0,0,0,0],
                  [0,0,1/2,1/3,0,0,1/3,0],
                   [0,0,0,1/3,1/3,0,0,1/2],
                   [0,0,0,0,1/3,0,0,1/2],
                   [0,0,0,0,1/3,1,1/3,0]])

# Compute eigenvalues and eigencevectrs
eigenvalues, eigenvectors = np.linalg.eig(adj_mat)

# Eigen vector corresponding to 1
eigen_1 = abs(eigenvectors[:,0])/np.linalg.norm(eigenvectors[:,0],1)

# most important page
page = np.where(eigen_1 == eigen_1.max())[0][0]+1
print(page)

# Code ends here


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# Code starts here

# Initialize stationary vector I
init_I = [1,0,0,0,0,0,0,0]

# Perform iterations for power method
for i in range(10):
    init_I = np.dot(adj_mat,init_I)
power_page = np.argmax(init_I)+1
power_page

# Code ends here


# --------------
# Code starts here

# New Adjancency matrix
# New Adjancency matrix
new_adj_mat = np.array([[0,0,0,0,0,0,0,0],
                   [1/2,0,1/2,1/3,0,0,0,0],
                  [1/2,0,0,0,0,0,0,0],
                   [0,1,0,0,0,0,0,0],
                   [0,0,1/2,1/3,0,0,1/2,0],
                   [0,0,0,1/3,1/3,0,0,1/2],
                   [0,0,0,0,1/3,0,0,1/2],
                   [0,0,0,0,1/3,1,1/2,0]])

# Initialize stationary vector I
new_init_I = [1,0,0,0,0,0,0,0]

# Perform iterations for power method
for i in range(10):
    new_init_I = np.dot(new_adj_mat, new_init_I)
new_init_I

# Code ends here


# --------------
# Alpha value
alpha = 0.85

# Code starts here

# Modified adjancency matrix
n = len(new_adj_mat)
G = (alpha * new_adj_mat) + (((1 - alpha) * (1/n)) * np.ones(new_adj_mat.shape))

# Initialize stationary vector I
final_init_I = [1,0,0,0,0,0,0,0]

# Perform iterations for power method
for i in range(1000):
    final_init_I = np.dot(G,final_init_I)
    final_init_I /= np.linalg.norm(final_init_I,1)
final_init_I

# Code ends here


