"""Hubs and authorities analysis of graph structure."""
import networkx as nx
__all__ = ["hits"]
[docs]
@nx._dispatchable(preserve_edge_attrs={"G": {"weight": 1}})
def hits(G, max_iter=100, tol=1.0e-8, nstart=None, normalized=True):
"""Returns HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node.
Authorities estimates the node value based on the incoming links.
Hubs estimates the node value based on outgoing links.
Parameters
----------
G : graph
A NetworkX graph
max_iter : integer, optional
Maximum number of iterations in power method.
tol : float, optional
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of each node for power method iteration.
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns
-------
(hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority
values.
Raises
------
PowerIterationFailedConvergence
If the algorithm fails to converge to the specified tolerance
within the specified number of iterations of the power iteration
method.
Examples
--------
>>> G = nx.path_graph(4)
>>> h, a = nx.hits(G)
Notes
-----
The eigenvector calculation is done by the power iteration method
and has no guarantee of convergence. The iteration will stop
after max_iter iterations or an error tolerance of
number_of_nodes(G)*tol has been reached.
The HITS algorithm was designed for directed graphs but this
algorithm does not check if the input graph is directed and will
execute on undirected graphs.
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Jon Kleinberg,
Authoritative sources in a hyperlinked environment
Journal of the ACM 46 (5): 604-32, 1999.
doi:10.1145/324133.324140.
http://www.cs.cornell.edu/home/kleinber/auth.pdf.
"""
import numpy as np
import scipy as sp
if len(G) == 0:
return {}, {}
A = nx.adjacency_matrix(G, nodelist=list(G), dtype=float)
if nstart is not None:
nstart = np.array(list(nstart.values()))
if max_iter <= 0:
raise nx.PowerIterationFailedConvergence(max_iter)
try:
_, _, vt = sp.sparse.linalg.svds(A, k=1, v0=nstart, maxiter=max_iter, tol=tol)
except sp.sparse.linalg.ArpackNoConvergence as exc:
raise nx.PowerIterationFailedConvergence(max_iter) from exc
a = vt.flatten().real
h = A @ a
if normalized:
h /= h.sum()
a /= a.sum()
hubs = dict(zip(G, map(float, h)))
authorities = dict(zip(G, map(float, a)))
return hubs, authorities
def _hits_python(G, max_iter=100, tol=1.0e-8, nstart=None, normalized=True):
if isinstance(G, nx.MultiGraph | nx.MultiDiGraph):
raise Exception("hits() not defined for graphs with multiedges.")
if len(G) == 0:
return {}, {}
# choose fixed starting vector if not given
if nstart is None:
h = dict.fromkeys(G, 1.0 / G.number_of_nodes())
else:
h = nstart
# normalize starting vector
s = 1.0 / sum(h.values())
for k in h:
h[k] *= s
for _ in range(max_iter): # power iteration: make up to max_iter iterations
hlast = h
h = dict.fromkeys(hlast.keys(), 0)
a = dict.fromkeys(hlast.keys(), 0)
# this "matrix multiply" looks odd because it is
# doing a left multiply a^T=hlast^T*G
for n in h:
for nbr in G[n]:
a[nbr] += hlast[n] * G[n][nbr].get("weight", 1)
# now multiply h=Ga
for n in h:
for nbr in G[n]:
h[n] += a[nbr] * G[n][nbr].get("weight", 1)
# normalize vector
s = 1.0 / max(h.values())
for n in h:
h[n] *= s
# normalize vector
s = 1.0 / max(a.values())
for n in a:
a[n] *= s
# check convergence, l1 norm
err = sum(abs(h[n] - hlast[n]) for n in h)
if err < tol:
break
else:
raise nx.PowerIterationFailedConvergence(max_iter)
if normalized:
s = 1.0 / sum(a.values())
for n in a:
a[n] *= s
s = 1.0 / sum(h.values())
for n in h:
h[n] *= s
return h, a
def _hits_numpy(G, normalized=True):
"""Returns HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node.
Authorities estimates the node value based on the incoming links.
Hubs estimates the node value based on outgoing links.
Parameters
----------
G : graph
A NetworkX graph
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns
-------
(hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority
values.
Examples
--------
>>> G = nx.path_graph(4)
The `hubs` and `authorities` are given by the eigenvectors corresponding to the
maximum eigenvalues of the hubs_matrix and the authority_matrix, respectively.
The ``hubs`` and ``authority`` matrices are computed from the adjacency
matrix:
>>> adj_ary = nx.to_numpy_array(G)
>>> hubs_matrix = adj_ary @ adj_ary.T
>>> authority_matrix = adj_ary.T @ adj_ary
`_hits_numpy` maps the eigenvector corresponding to the maximum eigenvalue
of the respective matrices to the nodes in `G`:
>>> from networkx.algorithms.link_analysis.hits_alg import _hits_numpy
>>> hubs, authority = _hits_numpy(G)
Notes
-----
The eigenvector calculation uses NumPy's interface to LAPACK.
The HITS algorithm was designed for directed graphs but this
algorithm does not check if the input graph is directed and will
execute on undirected graphs.
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Jon Kleinberg,
Authoritative sources in a hyperlinked environment
Journal of the ACM 46 (5): 604-32, 1999.
doi:10.1145/324133.324140.
http://www.cs.cornell.edu/home/kleinber/auth.pdf.
"""
import numpy as np
if len(G) == 0:
return {}, {}
adj_ary = nx.to_numpy_array(G)
# Hub matrix
H = adj_ary @ adj_ary.T
e, ev = np.linalg.eig(H)
h = ev[:, np.argmax(e)] # eigenvector corresponding to the maximum eigenvalue
# Authority matrix
A = adj_ary.T @ adj_ary
e, ev = np.linalg.eig(A)
a = ev[:, np.argmax(e)] # eigenvector corresponding to the maximum eigenvalue
if normalized:
h /= h.sum()
a /= a.sum()
else:
h /= h.max()
a /= a.max()
hubs = dict(zip(G, map(float, h)))
authorities = dict(zip(G, map(float, a)))
return hubs, authorities
def _hits_scipy(G, max_iter=100, tol=1.0e-6, nstart=None, normalized=True):
"""Returns HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node.
Authorities estimates the node value based on the incoming links.
Hubs estimates the node value based on outgoing links.
Parameters
----------
G : graph
A NetworkX graph
max_iter : integer, optional
Maximum number of iterations in power method.
tol : float, optional
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of each node for power method iteration.
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns
-------
(hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority
values.
Examples
--------
>>> from networkx.algorithms.link_analysis.hits_alg import _hits_scipy
>>> G = nx.path_graph(4)
>>> h, a = _hits_scipy(G)
Notes
-----
This implementation uses SciPy sparse matrices.
The eigenvector calculation is done by the power iteration method
and has no guarantee of convergence. The iteration will stop
after max_iter iterations or an error tolerance of
number_of_nodes(G)*tol has been reached.
The HITS algorithm was designed for directed graphs but this
algorithm does not check if the input graph is directed and will
execute on undirected graphs.
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] Jon Kleinberg,
Authoritative sources in a hyperlinked environment
Journal of the ACM 46 (5): 604-632, 1999.
doi:10.1145/324133.324140.
http://www.cs.cornell.edu/home/kleinber/auth.pdf.
"""
import numpy as np
if len(G) == 0:
return {}, {}
A = nx.to_scipy_sparse_array(G, nodelist=list(G))
(n, _) = A.shape # should be square
ATA = A.T @ A # authority matrix
# choose fixed starting vector if not given
if nstart is None:
x = np.ones((n, 1)) / n
else:
x = np.array([nstart.get(n, 0) for n in list(G)], dtype=float)
x /= x.sum()
# power iteration on authority matrix
i = 0
while True:
xlast = x
x = ATA @ x
x /= x.max()
# check convergence, l1 norm
err = np.absolute(x - xlast).sum()
if err < tol:
break
if i > max_iter:
raise nx.PowerIterationFailedConvergence(max_iter)
i += 1
a = x.flatten()
h = A @ a
if normalized:
h /= h.sum()
a /= a.sum()
hubs = dict(zip(G, map(float, h)))
authorities = dict(zip(G, map(float, a)))
return hubs, authorities
