@ -52,6 +52,8 @@ def load_data(dataset: Literal["Stanford", "BerkStan"]) -> nx.Graph:
# create the graph
print ( f " \n Creating the graph of the dataset { dataset } ... \n " )
G_dataset = nx . read_edgelist ( f " data/Web- { dataset } .txt " , create_using = nx . DiGraph ( ) , nodetype = int )
print ( f " \t Number of nodes: { G_dataset . number_of_nodes ( ) } " )
print ( f " \t Number of edges: { G_dataset . number_of_edges ( ) } " )
return G_dataset
@ -202,7 +204,7 @@ def pagerank_numpy(G, alpha=0.85, personalization=None, weight="weight", danglin
norm = largest . sum ( )
return dict ( zip ( G , map ( float , largest / norm ) ) )
def pagerank ( G , alpha = 0.85 , personalization = None , max_iter = 1 00, tol = 1.0e-9 , nstart = None , weight = " weight " , dangling = None , ) :
def pagerank ( G , alpha = 0.85 , personalization = None , max_iter = 2 00, tol = 1.0e-9 , nstart = None , weight = " weight " , dangling = None , ) :
"""
Returns the PageRank of the nodes in the graph .
@ -310,15 +312,52 @@ def pagerank(G, alpha=0.85, personalization=None, max_iter=100, tol=1.0e-9, nsta
# check convergence, l1 norm
err = np . absolute ( x - xlast ) . sum ( ) # err is the error between the current and previous vectors of PageRank values
if err < N * tol : # if the error is small enough, stop iterating
return dict ( zip ( nodelist , map ( float , x ) ) ) , iter , alpha , tol # return the current vector of PageRank values
raise nx . PowerIterationFailedConvergence ( max_iter ) # if the error is not small enough, raise an error
return dict ( zip ( nodelist , map ( float , x ) ) ) , iter , tol # return the current vector of PageRank values'
# other wise, return a Null dictionary, the number of iterations, and the tolerance
# this is a failure to convergeS
return { } , iter , tol
def shifted_pow_pagerank ( G , alphas = [ 0.85 , 0.9 , 0.95 , 0.99 ] , max_iter = 200 , tol = 1.0e-9 ) :
"""
Compute the PageRank of each node in the graph G .
Parameters
- - - - - - - - - -
G : graph
A NetworkX graph . Undirected graphs will be converted to a directed graph .
alphas : list , optional
A list of alpha values to use in the shifted power method . The default is [ 0.85 , 0.9 , 0.95 , 0.99 ] .
max_iter : integer , optional
Maximum number of iterations in power method eigenvalue solver .
tol : float , optional
Error tolerance used to check convergence in power method solver .
def shifted_pow_pagerank ( G , alphas = [ 0.85 , 0.9 , 0.95 , 0.99 ] , max_iter = 100 , tol = 1.0e-9 ) :
Returns
- - - - - - -
pagerank : dictionary
Dictionary of nodes with PageRank as value
mv : integer
The number of matrix - vector multiplications used in the power method
Notes
- - - - -
The eigenvector calculation uses power iteration with a SciPy sparse matrix representation . The shifted power method is described as Algorithm 1 in the paper located in the sources folders .
"""
N = len ( G )
if N == 0 :
return { }
# initialize a random sparse matrix of dimension N x len(alphas). The cols of this matrix are the page rank vectors for each alpha.
x = sp . sparse . random ( N , len ( alphas ) , density = 0.01 , format = " lil " , dtype = float )
nodelist = list ( G )
A = nx . to_scipy_sparse_array ( G , nodelist = nodelist , dtype = float )
@ -327,34 +366,30 @@ def shifted_pow_pagerank(G, alphas=[0.85, 0.9, 0.95, 0.99], max_iter=100, tol=1.
Q = sp . sparse . csr_array ( sp . sparse . spdiags ( S . T , 0 , * A . shape ) ) # Q is the matrix of edge weights going into each node
A = Q
x = np . repeat ( 1.0 / N , N ) # x is the vector of PageRank values
v = np . repeat ( 1.0 / N , N ) # p is the personalization vector
mu = A @ v - v # mu is the vector of PageRank values for the random walk with restart
mu = A @ v - v
for i in range ( len ( alphas ) ) :
# create a vector r of len(alphas) where r[i] = alpha[i] * mu
r = alphas [ i ] * mu
Res = np . linalg . norm ( r , 2 )
r = alphas [ i ] * mu # residual vector
Res = np . linalg . norm ( r , 2 ) # residual norm
if Res > = tol :
x = r + v # update x
x [ : , [ i ] ] = r + v # update the i-th column of x
iter = 1
mv = 0 # number of matrix-vector multiplications
for _ in range ( max_iter ) :
xlast = x
iter + = 1
mu = A @ x - x
mv + = 1
mu = A @ mu
for i in range ( len ( alphas ) ) :
r = alphas [ i ] * * ( iter + 1 ) * mu
Res = np . linalg . norm ( r , 2 )
if Res > = tol :
x = r + x
r = pow ( alphas [ i ] , mv + 1 ) * mu
Res = np . linalg . norm ( r , 2 )
err = np . absolute ( x - xlast ) . sum ( ) # err is the error between the current and previous vectors of PageRank values
if Res > = tol :
x [ : , [ i ] ] = r + v
if err < tol :
return dict ( zip ( nodelist , map ( float , x ) ) ) , iter , alphas , tol
err = np . absolute ( r ) . max ( )
if err < tol :
return x , mv , alphas , tol
raise nx . PowerIterationFailedConvergence ( max_iter ) # if the error is not small enough, raise an error