"""PageRank analysis of graph structure."""
from warnings import warn
import networkx as nx
__all__ = ["pagerank", "google_matrix"]
def _pagerank_python(
G,
alpha=0.85,
personalization=None,
max_iter=100,
tol=1.0e-6,
nstart=None,
weight="weight",
dangling=None,
):
if len(G) == 0:
return {}
D = G.to_directed()
# Create a copy in (right) stochastic form
W = nx.stochastic_graph(D, weight=weight)
N = W.number_of_nodes()
# Choose fixed starting vector if not given
if nstart is None:
x = dict.fromkeys(W, 1.0 / N)
else:
# Normalized nstart vector
s = sum(nstart.values())
x = {k: v / s for k, v in nstart.items()}
if personalization is None:
# Assign uniform personalization vector if not given
p = dict.fromkeys(W, 1.0 / N)
else:
s = sum(personalization.values())
p = {k: v / s for k, v in personalization.items()}
if dangling is None:
# Use personalization vector if dangling vector not specified
dangling_weights = p
else:
s = sum(dangling.values())
dangling_weights = {k: v / s for k, v in dangling.items()}
dangling_nodes = [n for n in W if W.out_degree(n, weight=weight) == 0.0]
# power iteration: make up to max_iter iterations
for _ in range(max_iter):
xlast = x
x = dict.fromkeys(xlast.keys(), 0)
danglesum = alpha * sum(xlast[n] for n in dangling_nodes)
for n in x:
# this matrix multiply looks odd because it is
# doing a left multiply x^T=xlast^T*W
for _, nbr, wt in W.edges(n, data=weight):
x[nbr] += alpha * xlast[n] * wt
x[n] += danglesum * dangling_weights.get(n, 0) + (1.0 - alpha) * p.get(n, 0)
# check convergence, l1 norm
err = sum(abs(x[n] - xlast[n]) for n in x)
if err < N * tol:
return x
raise nx.PowerIterationFailedConvergence(max_iter)
[docs]
@nx._dispatchable(edge_attrs="weight")
def google_matrix(
G, alpha=0.85, personalization=None, nodelist=None, weight="weight", dangling=None
):
"""Returns the Google matrix of the graph.
Parameters
----------
G : graph
A NetworkX graph. Undirected graphs will be converted to a directed
graph with two directed edges for each undirected edge.
alpha : float
The damping factor.
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key some subset of graph nodes and personalization value each of those.
At least one personalization value must be non-zero.
If not specified, a nodes personalization value will be zero.
By default, a uniform distribution is used.
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist.
If nodelist is None, then the ordering is produced by G.nodes().
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any "dangling" nodes, i.e., nodes without
any outedges. The dict key is the node the outedge points to and the dict
value is the weight of that outedge. By default, dangling nodes are given
outedges according to the personalization vector (uniform if not
specified) This must be selected to result in an irreducible transition
matrix (see notes below). It may be common to have the dangling dict to
be the same as the personalization dict.
Returns
-------
A : 2D NumPy ndarray
Google matrix of the graph
Notes
-----
The array returned represents the transition matrix that describes the
Markov chain used in PageRank. For PageRank to converge to a unique
solution (i.e., a unique stationary distribution in a Markov chain), the
transition matrix must be irreducible. In other words, it must be that
there exists a path between every pair of nodes in the graph, or else there
is the potential of "rank sinks."
This implementation works with Multi(Di)Graphs. For multigraphs the
weight between two nodes is set to be the sum of all edge weights
between those nodes.
See Also
--------
pagerank
"""
import numpy as np
if nodelist is None:
nodelist = list(G)
A = nx.to_numpy_array(G, nodelist=nodelist, weight=weight)
N = len(G)
if N == 0:
return A
# Personalization vector
if personalization is None:
p = np.repeat(1.0 / N, N)
else:
p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float)
if p.sum() == 0:
raise ZeroDivisionError
p /= p.sum()
# Dangling nodes
if dangling is None:
dangling_weights = p
else:
# Convert the dangling dictionary into an array in nodelist order
dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float)
dangling_weights /= dangling_weights.sum()
dangling_nodes = np.where(A.sum(axis=1) == 0)[0]
# Assign dangling_weights to any dangling nodes (nodes with no out links)
A[dangling_nodes] = dangling_weights
A /= A.sum(axis=1)[:, np.newaxis] # Normalize rows to sum to 1
return alpha * A + (1 - alpha) * p
def _pagerank_numpy(
G, alpha=0.85, personalization=None, weight="weight", dangling=None
):
"""Returns the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on
the structure of the incoming links. It was originally designed as
an algorithm to rank web pages.
Parameters
----------
G : graph
A NetworkX graph. Undirected graphs will be converted to a directed
graph with two directed edges for each undirected edge.
alpha : float, optional
Damping parameter for PageRank, default=0.85.
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key some subset of graph nodes and personalization value each of those.
At least one personalization value must be non-zero.
If not specified, a nodes personalization value will be zero.
By default, a uniform distribution is used.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any "dangling" nodes, i.e., nodes without
any outedges. The dict key is the node the outedge points to and the dict
value is the weight of that outedge. By default, dangling nodes are given
outedges according to the personalization vector (uniform if not
specified) This must be selected to result in an irreducible transition
matrix (see notes under google_matrix). It may be common to have the
dangling dict to be the same as the personalization dict.
Returns
-------
pagerank : dictionary
Dictionary of nodes with PageRank as value.
Examples
--------
>>> from networkx.algorithms.link_analysis.pagerank_alg import _pagerank_numpy
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = _pagerank_numpy(G, alpha=0.9)
Notes
-----
The eigenvector calculation uses NumPy's interface to the LAPACK
eigenvalue solvers. This will be the fastest and most accurate
for small graphs.
This implementation works with Multi(Di)Graphs. For multigraphs the
weight between two nodes is set to be the sum of all edge weights
between those nodes.
See Also
--------
pagerank, google_matrix
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry,
The PageRank citation ranking: Bringing order to the Web. 1999
http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf
"""
import numpy as np
if len(G) == 0:
return {}
M = google_matrix(
G, alpha, personalization=personalization, weight=weight, dangling=dangling
)
# use numpy LAPACK solver
eigenvalues, eigenvectors = np.linalg.eig(M.T)
ind = np.argmax(eigenvalues)
# eigenvector of largest eigenvalue is at ind, normalized
largest = np.array(eigenvectors[:, ind]).flatten().real
norm = largest.sum()
return dict(zip(G, map(float, largest / norm)))
def _pagerank_scipy(
G,
alpha=0.85,
personalization=None,
max_iter=100,
tol=1.0e-6,
nstart=None,
weight="weight",
dangling=None,
):
"""Returns the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on
the structure of the incoming links. It was originally designed as
an algorithm to rank web pages.
Parameters
----------
G : graph
A NetworkX graph. Undirected graphs will be converted to a directed
graph with two directed edges for each undirected edge.
alpha : float, optional
Damping parameter for PageRank, default=0.85.
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key some subset of graph nodes and personalization value each of those.
At least one personalization value must be non-zero.
If not specified, a nodes personalization value will be zero.
By default, a uniform distribution is used.
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.
The iteration will stop after a tolerance of ``len(G) * tol`` is reached.
nstart : dictionary, optional
Starting value of PageRank iteration for each node.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any "dangling" nodes, i.e., nodes without
any outedges. The dict key is the node the outedge points to and the dict
value is the weight of that outedge. By default, dangling nodes are given
outedges according to the personalization vector (uniform if not
specified) This must be selected to result in an irreducible transition
matrix (see notes under google_matrix). It may be common to have the
dangling dict to be the same as the personalization dict.
Returns
-------
pagerank : dictionary
Dictionary of nodes with PageRank as value
Examples
--------
>>> from networkx.algorithms.link_analysis.pagerank_alg import _pagerank_scipy
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = _pagerank_scipy(G, alpha=0.9)
Notes
-----
The eigenvector calculation uses power iteration with a SciPy
sparse matrix representation.
This implementation works with Multi(Di)Graphs. For multigraphs the
weight between two nodes is set to be the sum of all edge weights
between those nodes.
See Also
--------
pagerank
Raises
------
PowerIterationFailedConvergence
If the algorithm fails to converge to the specified tolerance
within the specified number of iterations of the power iteration
method.
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry,
The PageRank citation ranking: Bringing order to the Web. 1999
http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf
"""
import numpy as np
import scipy as sp
N = len(G)
if N == 0:
return {}
nodelist = list(G)
A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, dtype=float)
S = A.sum(axis=1)
S[S != 0] = 1.0 / S[S != 0]
# TODO: csr_array
Q = sp.sparse.csr_array(sp.sparse.spdiags(S.T, 0, *A.shape))
A = Q @ A
# initial vector
if nstart is None:
x = np.repeat(1.0 / N, N)
else:
x = np.array([nstart.get(n, 0) for n in nodelist], dtype=float)
x /= x.sum()
# Personalization vector
if personalization is None:
p = np.repeat(1.0 / N, N)
else:
p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float)
if p.sum() == 0:
raise ZeroDivisionError
p /= p.sum()
# Dangling nodes
if dangling is None:
dangling_weights = p
else:
# Convert the dangling dictionary into an array in nodelist order
dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float)
dangling_weights /= dangling_weights.sum()
is_dangling = np.where(S == 0)[0]
# power iteration: make up to max_iter iterations
for _ in range(max_iter):
xlast = x
x = alpha * (x @ A + sum(x[is_dangling]) * dangling_weights) + (1 - alpha) * p
# check convergence, l1 norm
err = np.absolute(x - xlast).sum()
if err < N * tol:
return dict(zip(nodelist, map(float, x)))
raise nx.PowerIterationFailedConvergence(max_iter)
