"""Katz centrality."""
import math
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["katz_centrality", "katz_centrality_numpy"]
[docs]@not_implemented_for("multigraph")
def katz_centrality(
G,
alpha=0.1,
beta=1.0,
max_iter=1000,
tol=1.0e-6,
nstart=None,
normalized=True,
weight=None,
):
r"""Compute the Katz centrality for the nodes of the graph G.
Katz centrality computes the centrality for a node based on the centrality
of its neighbors. It is a generalization of the eigenvector centrality. The
Katz centrality for node $i$ is
.. math::
x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.
The parameter $\beta$ controls the initial centrality and
.. math::
\alpha < \frac{1}{\lambda_{\max}}.
Katz centrality computes the relative influence of a node within a
network by measuring the number of the immediate neighbors (first
degree nodes) and also all other nodes in the network that connect
to the node under consideration through these immediate neighbors.
Extra weight can be provided to immediate neighbors through the
parameter $\beta$. Connections made with distant neighbors
are, however, penalized by an attenuation factor $\alpha$ which
should be strictly less than the inverse largest eigenvalue of the
adjacency matrix in order for the Katz centrality to be computed
correctly. More information is provided in [1]_.
Parameters
----------
G : graph
A NetworkX graph.
alpha : float
Attenuation factor
beta : scalar or dictionary, optional (default=1.0)
Weight attributed to the immediate neighborhood. If not a scalar, the
dictionary must have an value for every node.
max_iter : integer, optional (default=1000)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of Katz iteration for each node.
normalized : bool, optional (default=True)
If True normalize the resulting values.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
In this measure the weight is interpreted as the connection strength.
Returns
-------
nodes : dictionary
Dictionary of nodes with Katz centrality as the value.
Raises
------
NetworkXError
If the parameter `beta` is not a scalar but lacks a value for at least
one node
PowerIterationFailedConvergence
If the algorithm fails to converge to the specified tolerance
within the specified number of iterations of the power iteration
method.
Examples
--------
>>> import math
>>> G = nx.path_graph(4)
>>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix
>>> centrality = nx.katz_centrality(G, 1 / phi - 0.01)
>>> for n, c in sorted(centrality.items()):
... print(f"{n} {c:.2f}")
0 0.37
1 0.60
2 0.60
3 0.37
See Also
--------
katz_centrality_numpy
eigenvector_centrality
eigenvector_centrality_numpy
pagerank
hits
Notes
-----
Katz centrality was introduced by [2]_.
This algorithm it uses the power method to find the eigenvector
corresponding to the largest eigenvalue of the adjacency matrix of ``G``.
The parameter ``alpha`` should be strictly less than the inverse of largest
eigenvalue of the adjacency matrix for the algorithm to converge.
You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
eigenvalue of the adjacency matrix.
The iteration will stop after ``max_iter`` iterations or an error tolerance of
``number_of_nodes(G) * tol`` has been reached.
When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same
as eigenvector centrality.
For directed graphs this finds "left" eigenvectors which corresponds
to the in-edges in the graph. For out-edges Katz centrality
first reverse the graph with ``G.reverse()``.
References
----------
.. [1] Mark E. J. Newman:
Networks: An Introduction.
Oxford University Press, USA, 2010, p. 720.
.. [2] Leo Katz:
A New Status Index Derived from Sociometric Index.
Psychometrika 18(1):39–43, 1953
https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
"""
if len(G) == 0:
return {}
nnodes = G.number_of_nodes()
if nstart is None:
# choose starting vector with entries of 0
x = {n: 0 for n in G}
else:
x = nstart
try:
b = dict.fromkeys(G, float(beta))
except (TypeError, ValueError, AttributeError) as err:
b = beta
if set(beta) != set(G):
raise nx.NetworkXError(
"beta dictionary " "must have a value for every node"
) from err
# make up to max_iter iterations
for i in range(max_iter):
xlast = x
x = dict.fromkeys(xlast, 0)
# do the multiplication y^T = Alpha * x^T A - Beta
for n in x:
for nbr in G[n]:
x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
for n in x:
x[n] = alpha * x[n] + b[n]
# check convergence
error = sum(abs(x[n] - xlast[n]) for n in x)
if error < nnodes * tol:
if normalized:
# normalize vector
try:
s = 1.0 / math.hypot(*x.values())
# this should never be zero?
except ZeroDivisionError:
s = 1.0
else:
s = 1
for n in x:
x[n] *= s
return x
raise nx.PowerIterationFailedConvergence(max_iter)
[docs]@not_implemented_for("multigraph")
def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None):
r"""Compute the Katz centrality for the graph G.
Katz centrality computes the centrality for a node based on the centrality
of its neighbors. It is a generalization of the eigenvector centrality. The
Katz centrality for node $i$ is
.. math::
x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.
The parameter $\beta$ controls the initial centrality and
.. math::
\alpha < \frac{1}{\lambda_{\max}}.
Katz centrality computes the relative influence of a node within a
network by measuring the number of the immediate neighbors (first
degree nodes) and also all other nodes in the network that connect
to the node under consideration through these immediate neighbors.
Extra weight can be provided to immediate neighbors through the
parameter $\beta$. Connections made with distant neighbors
are, however, penalized by an attenuation factor $\alpha$ which
should be strictly less than the inverse largest eigenvalue of the
adjacency matrix in order for the Katz centrality to be computed
correctly. More information is provided in [1]_.
Parameters
----------
G : graph
A NetworkX graph
alpha : float
Attenuation factor
beta : scalar or dictionary, optional (default=1.0)
Weight attributed to the immediate neighborhood. If not a scalar the
dictionary must have an value for every node.
normalized : bool
If True normalize the resulting values.
weight : None or string, optional
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
In this measure the weight is interpreted as the connection strength.
Returns
-------
nodes : dictionary
Dictionary of nodes with Katz centrality as the value.
Raises
------
NetworkXError
If the parameter `beta` is not a scalar but lacks a value for at least
one node
Examples
--------
>>> import math
>>> G = nx.path_graph(4)
>>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix
>>> centrality = nx.katz_centrality_numpy(G, 1 / phi)
>>> for n, c in sorted(centrality.items()):
... print(f"{n} {c:.2f}")
0 0.37
1 0.60
2 0.60
3 0.37
See Also
--------
katz_centrality
eigenvector_centrality_numpy
eigenvector_centrality
pagerank
hits
Notes
-----
Katz centrality was introduced by [2]_.
This algorithm uses a direct linear solver to solve the above equation.
The parameter ``alpha`` should be strictly less than the inverse of largest
eigenvalue of the adjacency matrix for there to be a solution.
You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
eigenvalue of the adjacency matrix.
When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same
as eigenvector centrality.
For directed graphs this finds "left" eigenvectors which corresponds
to the in-edges in the graph. For out-edges Katz centrality
first reverse the graph with ``G.reverse()``.
References
----------
.. [1] Mark E. J. Newman:
Networks: An Introduction.
Oxford University Press, USA, 2010, p. 173.
.. [2] Leo Katz:
A New Status Index Derived from Sociometric Index.
Psychometrika 18(1):39–43, 1953
https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
"""
import numpy as np
if len(G) == 0:
return {}
try:
nodelist = beta.keys()
if set(nodelist) != set(G):
raise nx.NetworkXError(
"beta dictionary " "must have a value for every node"
)
b = np.array(list(beta.values()), dtype=float)
except AttributeError:
nodelist = list(G)
try:
b = np.ones((len(nodelist), 1)) * float(beta)
except (TypeError, ValueError, AttributeError) as err:
raise nx.NetworkXError("beta must be a number") from err
A = nx.adjacency_matrix(G, nodelist=nodelist, weight=weight).todense().T
n = A.shape[0]
centrality = np.linalg.solve(np.eye(n, n) - (alpha * A), b)
if normalized:
norm = np.sign(sum(centrality)) * np.linalg.norm(centrality)
else:
norm = 1.0
centrality = dict(zip(nodelist, map(float, centrality / norm)))
return centrality