# --------------
# 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
print("page:")
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 _ in range (10):
    init_I = np.dot(adj_mat, 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


# --------------
# 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 _ 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)




# Code ends here


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

# Code starts here

# 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(G, final_init_I)
    final_init_I /= np.linalg.norm(final_init_I, 1)

print(final_init_I)
# Code ends here


