Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

"""PageRank analysis of graph structure. """
import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["pagerank", "pagerank_numpy", "pagerank_scipy", "google_matrix"]

[docs]@not_implemented_for("multigraph")
def pagerank(
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 specfiied, 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.

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
--------
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.pagerank(G, alpha=0.9)

Notes
-----
The eigenvector calculation is done by the power iteration method
and has no guarantee of convergence.  The iteration will stop after
an error tolerance of len(G) * tol has been reached. If the
number of iterations exceed max_iter, a
:exc:networkx.exception.PowerIterationFailedConvergence exception
is raised.

The PageRank algorithm was designed for directed graphs but this
algorithm does not check if the input graph is directed and will
execute on undirected graphs by converting each edge in the
directed graph to two edges.

--------

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

"""
if len(G) == 0:
return {}

if not G.is_directed():
D = G.to_directed()
else:
D = G

# 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 = float(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 = float(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 = float(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 in W[n]:
x[nbr] += alpha * xlast[n] * W[n][nbr][weight]
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)

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 specfiied, 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 : NumPy matrix

Notes
-----
The matrix 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.

--------
pagerank, pagerank_numpy, pagerank_scipy
"""
import numpy as np

if nodelist is None:
nodelist = list(G)

M = nx.to_numpy_matrix(G, nodelist=nodelist, weight=weight)
N = len(G)
if N == 0:
return M

# 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)
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(M.sum(axis=1) == 0)[0]

# Assign dangling_weights to any dangling nodes (nodes with no out links)
for node in dangling_nodes:
M[node] = dangling_weights

M /= M.sum(axis=1)  # Normalize rows to sum to 1

return alpha * M + (1 - alpha) * p

[docs]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 specfiied, 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
--------
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.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.

--------

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 {}
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 = float(largest.sum())
return dict(zip(G, map(float, largest / norm)))

[docs]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 specfiied, 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.

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
--------
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.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.

--------

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.sparse

N = len(G)
if N == 0:
return {}

nodelist = list(G)
M = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, dtype=float)
S = np.array(M.sum(axis=1)).flatten()
S[S != 0] = 1.0 / S[S != 0]
Q = scipy.sparse.spdiags(S.T, 0, *M.shape, format="csr")
M = Q * M

# 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 / 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)
p = 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 * M + 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)