# --------------
# 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 ==np.max(eigen_1))[0][0]+1
print(page)

# Code ends here


# --------------
# Code starts here
import numpy as np
# Initialize stationary vector I
init_I = np.array([1,0,0,0,0,0,0,0])

# Perform iterations for power method
for i in range(0,10):
  init_I = np.dot(adj_mat,init_I)/np.linalg.norm(init_I,1)

    
power_page = np.where(np.max(init_I) == init_I)[0][0]+1
print(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 = np.array([1,0,0,0,0,0,0,0])

# Perform iterations for power method
for i in range(0,10):
    new_init_I = np.dot(new_adj_mat, new_init_I)/np.linalg.norm(new_init_I,1)

print(new_init_I)


# Code ends here


# --------------
# Alpha value
alpha = 0.85
from scipy.linalg import svd


# Modified adjancency matrix
u,S,vt = svd(new_adj_mat)
G = alpha*S + np.ones(new_adj_mat.shape)-alpha/len(new_adj_mat)

# Initialize stationary vector I
final_init_I = np.array([1,0,0,0,0,0,0,0])

# Perform iterations for power method
for i in range(0,1000):
    final_init_I = np.dot(new_adj_mat,final_init_I)/np.linalg.norm(final_init_I,1)
print(final_init_I)
# Code ends here


