# --------------
# 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
abs(eigenvectors[:,0])
eigen_1 = abs(eigenvectors[:,0])/np.linalg.norm(eigenvectors[:,0],1)

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

# Code ends here
print(page)




# --------------
# 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 _ in range(10):
    init_I = np.dot(adj_mat,init_I)
    init_I = init_I/np.linalg.norm(init_I,1)


print(init_I)
power_page = np.where(np.max(init_I) == init_I)[0][0]+1
print(power_page)
# Code ends here


# --------------
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]])
                   
new_init_I = np.array([1,0,0,0,0,0,0,0])
for _ in range(10):
    new_init_I  = np.dot(new_adj_mat, new_init_I)
    new_init_I /= np.linalg.norm(new_init_I,1)

print(new_init_I)



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

# Code starts here

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]])

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

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

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


print(final_init_I)
# Code ends here


