"""
Subraph centrality and communicability betweenness.
"""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"subgraph_centrality_exp",
"subgraph_centrality",
"communicability_betweenness_centrality",
"estrada_index",
]
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def subgraph_centrality_exp(G):
r"""Returns the subgraph centrality for each node of G.
Subgraph centrality of a node `n` is the sum of weighted closed
walks of all lengths starting and ending at node `n`. The weights
decrease with path length. Each closed walk is associated with a
connected subgraph ([1]_).
Parameters
----------
G: graph
Returns
-------
nodes:dictionary
Dictionary of nodes with subgraph centrality as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
subgraph_centrality:
Alternative algorithm of the subgraph centrality for each node of G.
Notes
-----
This version of the algorithm exponentiates the adjacency matrix.
The subgraph centrality of a node `u` in G can be found using
the matrix exponential of the adjacency matrix of G [1]_,
.. math::
SC(u)=(e^A)_{uu} .
References
----------
.. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
"Subgraph centrality in complex networks",
Physical Review E 71, 056103 (2005).
https://arxiv.org/abs/cond-mat/0504730
Examples
--------
(Example from [1]_)
>>> G = nx.Graph(
... [
... (1, 2),
... (1, 5),
... (1, 8),
... (2, 3),
... (2, 8),
... (3, 4),
... (3, 6),
... (4, 5),
... (4, 7),
... (5, 6),
... (6, 7),
... (7, 8),
... ]
... )
>>> sc = nx.subgraph_centrality_exp(G)
>>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)])
['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
"""
# alternative implementation that calculates the matrix exponential
import scipy as sp
nodelist = list(G) # ordering of nodes in matrix
A = nx.to_numpy_array(G, nodelist)
# convert to 0-1 matrix
A[A != 0.0] = 1
expA = sp.linalg.expm(A)
# convert diagonal to dictionary keyed by node
sc = dict(zip(nodelist, map(float, expA.diagonal())))
return sc
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def subgraph_centrality(G):
r"""Returns subgraph centrality for each node in G.
Subgraph centrality of a node `n` is the sum of weighted closed
walks of all lengths starting and ending at node `n`. The weights
decrease with path length. Each closed walk is associated with a
connected subgraph ([1]_).
Parameters
----------
G: graph
Returns
-------
nodes : dictionary
Dictionary of nodes with subgraph centrality as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
subgraph_centrality_exp:
Alternative algorithm of the subgraph centrality for each node of G.
Notes
-----
This version of the algorithm computes eigenvalues and eigenvectors
of the adjacency matrix.
Subgraph centrality of a node `u` in G can be found using
a spectral decomposition of the adjacency matrix [1]_,
.. math::
SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},
where `v_j` is an eigenvector of the adjacency matrix `A` of G
corresponding to the eigenvalue `\lambda_j`.
Examples
--------
(Example from [1]_)
>>> G = nx.Graph(
... [
... (1, 2),
... (1, 5),
... (1, 8),
... (2, 3),
... (2, 8),
... (3, 4),
... (3, 6),
... (4, 5),
... (4, 7),
... (5, 6),
... (6, 7),
... (7, 8),
... ]
... )
>>> sc = nx.subgraph_centrality(G)
>>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)])
['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
References
----------
.. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
"Subgraph centrality in complex networks",
Physical Review E 71, 056103 (2005).
https://arxiv.org/abs/cond-mat/0504730
"""
import numpy as np
nodelist = list(G) # ordering of nodes in matrix
A = nx.to_numpy_array(G, nodelist)
# convert to 0-1 matrix
A[np.nonzero(A)] = 1
w, v = np.linalg.eigh(A)
vsquare = np.array(v) ** 2
expw = np.exp(w)
xg = vsquare @ expw
# convert vector dictionary keyed by node
sc = dict(zip(nodelist, map(float, xg)))
return sc
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def communicability_betweenness_centrality(G):
r"""Returns subgraph communicability for all pairs of nodes in G.
Communicability betweenness measure makes use of the number of walks
connecting every pair of nodes as the basis of a betweenness centrality
measure.
Parameters
----------
G: graph
Returns
-------
nodes : dictionary
Dictionary of nodes with communicability betweenness as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
Notes
-----
Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges,
and `A` denote the adjacency matrix of `G`.
Let `G(r)=(V,E(r))` be the graph resulting from
removing all edges connected to node `r` but not the node itself.
The adjacency matrix for `G(r)` is `A+E(r)`, where `E(r)` has nonzeros
only in row and column `r`.
The subraph betweenness of a node `r` is [1]_
.. math::
\omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}},
p\neq q, q\neq r,
where
`G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}` is the number of walks
involving node r,
`G_{pq}=(e^{A})_{pq}` is the number of closed walks starting
at node `p` and ending at node `q`,
and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the
number of terms in the sum.
The resulting `\omega_{r}` takes values between zero and one.
The lower bound cannot be attained for a connected
graph, and the upper bound is attained in the star graph.
References
----------
.. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano,
"Communicability Betweenness in Complex Networks"
Physica A 388 (2009) 764-774.
https://arxiv.org/abs/0905.4102
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
>>> cbc = nx.communicability_betweenness_centrality(G)
>>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)])
['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03']
"""
import numpy as np
import scipy as sp
nodelist = list(G) # ordering of nodes in matrix
n = len(nodelist)
A = nx.to_numpy_array(G, nodelist)
# convert to 0-1 matrix
A[np.nonzero(A)] = 1
expA = sp.linalg.expm(A)
mapping = dict(zip(nodelist, range(n)))
cbc = {}
for v in G:
# remove row and col of node v
i = mapping[v]
row = A[i, :].copy()
col = A[:, i].copy()
A[i, :] = 0
A[:, i] = 0
B = (expA - sp.linalg.expm(A)) / expA
# sum with row/col of node v and diag set to zero
B[i, :] = 0
B[:, i] = 0
B -= np.diag(np.diag(B))
cbc[v] = float(B.sum())
# put row and col back
A[i, :] = row
A[:, i] = col
# rescale when more than two nodes
order = len(cbc)
if order > 2:
scale = 1.0 / ((order - 1.0) ** 2 - (order - 1.0))
cbc = {node: value * scale for node, value in cbc.items()}
return cbc
[docs]
@nx._dispatchable
def estrada_index(G):
r"""Returns the Estrada index of a the graph G.
The Estrada Index is a topological index of folding or 3D "compactness" ([1]_).
Parameters
----------
G: graph
Returns
-------
estrada index: float
Raises
------
NetworkXError
If the graph is not undirected and simple.
Notes
-----
Let `G=(V,E)` be a simple undirected graph with `n` nodes and let
`\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{n}`
be a non-increasing ordering of the eigenvalues of its adjacency
matrix `A`. The Estrada index is ([1]_, [2]_)
.. math::
EE(G)=\sum_{j=1}^n e^{\lambda _j}.
References
----------
.. [1] E. Estrada, "Characterization of 3D molecular structure",
Chem. Phys. Lett. 319, 713 (2000).
https://doi.org/10.1016/S0009-2614(00)00158-5
.. [2] José Antonio de la Peñaa, Ivan Gutman, Juan Rada,
"Estimating the Estrada index",
Linear Algebra and its Applications. 427, 1 (2007).
https://doi.org/10.1016/j.laa.2007.06.020
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
>>> ei = nx.estrada_index(G)
>>> print(f"{ei:0.5}")
20.55
"""
return sum(subgraph_centrality(G).values())
