# --------------
# 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
eigrnvalues , eigenvectors = np.linalg.eig(adj_mat)

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

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


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

# 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(10):
    init_I = np.dot(adj_mat,init_I)
    init_I = 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
import numpy as np
# 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(10):
    new_init_I = np.dot(new_adj_mat,new_init_I)
    new_init_I = new_init_I / np.linalg.norm(new_init_I,1)

#display
print (new_init_I)




# Code ends here


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

# Code starts here

# Modified adjancency matrix
n = len(new_adj_mat)
l = np.ones(new_adj_mat.shape)
G = alpha*new_adj_mat + (l-alpha)/n
# 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(1000):
    final_init_I = np.dot(G,final_init_I)
    final_init_I = final_init_I / np.linalg.norm(final_init_I,1)

#display
print(final_init_I)
# Code ends here


