608 lines
18 KiB
Python
608 lines
18 KiB
Python
|
"""Graph diameter, radius, eccentricity and other properties."""
|
||
|
|
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.utils import not_implemented_for
|
||
|
|
|
||
|
|
__all__ = [
|
||
|
|
"extrema_bounding",
|
||
|
|
"eccentricity",
|
||
|
|
"diameter",
|
||
|
|
"radius",
|
||
|
|
"periphery",
|
||
|
|
"center",
|
||
|
|
"barycenter",
|
||
|
|
"resistance_distance",
|
||
|
|
]
|
||
|
|
|
||
|
|
|
||
|
|
def extrema_bounding(G, compute="diameter"):
|
||
|
|
"""Compute requested extreme distance metric of undirected graph G
|
||
|
|
|
||
|
|
Computation is based on smart lower and upper bounds, and in practice
|
||
|
|
linear in the number of nodes, rather than quadratic (except for some
|
||
|
|
border cases such as complete graphs or circle shaped graphs).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
An undirected graph
|
||
|
|
|
||
|
|
compute : string denoting the requesting metric
|
||
|
|
"diameter" for the maximal eccentricity value,
|
||
|
|
"radius" for the minimal eccentricity value,
|
||
|
|
"periphery" for the set of nodes with eccentricity equal to the diameter
|
||
|
|
"center" for the set of nodes with eccentricity equal to the radius
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
value : value of the requested metric
|
||
|
|
int for "diameter" and "radius" or
|
||
|
|
list of nodes for "center" and "periphery"
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
NetworkXError
|
||
|
|
If the graph consists of multiple components
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This algorithm was proposed in the following papers:
|
||
|
|
|
||
|
|
F.W. Takes and W.A. Kosters, Determining the Diameter of Small World
|
||
|
|
Networks, in Proceedings of the 20th ACM International Conference on
|
||
|
|
Information and Knowledge Management (CIKM 2011), pp. 1191-1196, 2011.
|
||
|
|
doi: https://doi.org/10.1145/2063576.2063748
|
||
|
|
|
||
|
|
F.W. Takes and W.A. Kosters, Computing the Eccentricity Distribution of
|
||
|
|
Large Graphs, Algorithms 6(1): 100-118, 2013.
|
||
|
|
doi: https://doi.org/10.3390/a6010100
|
||
|
|
|
||
|
|
M. Borassi, P. Crescenzi, M. Habib, W.A. Kosters, A. Marino and F.W. Takes,
|
||
|
|
Fast Graph Diameter and Radius BFS-Based Computation in (Weakly Connected)
|
||
|
|
Real-World Graphs, Theoretical Computer Science 586: 59-80, 2015.
|
||
|
|
doi: https://doi.org/10.1016/j.tcs.2015.02.033
|
||
|
|
"""
|
||
|
|
|
||
|
|
# init variables
|
||
|
|
degrees = dict(G.degree()) # start with the highest degree node
|
||
|
|
minlowernode = max(degrees, key=degrees.get)
|
||
|
|
N = len(degrees) # number of nodes
|
||
|
|
# alternate between smallest lower and largest upper bound
|
||
|
|
high = False
|
||
|
|
# status variables
|
||
|
|
ecc_lower = dict.fromkeys(G, 0)
|
||
|
|
ecc_upper = dict.fromkeys(G, N)
|
||
|
|
candidates = set(G)
|
||
|
|
|
||
|
|
# (re)set bound extremes
|
||
|
|
minlower = N
|
||
|
|
maxlower = 0
|
||
|
|
minupper = N
|
||
|
|
maxupper = 0
|
||
|
|
|
||
|
|
# repeat the following until there are no more candidates
|
||
|
|
while candidates:
|
||
|
|
if high:
|
||
|
|
current = maxuppernode # select node with largest upper bound
|
||
|
|
else:
|
||
|
|
current = minlowernode # select node with smallest lower bound
|
||
|
|
high = not high
|
||
|
|
|
||
|
|
# get distances from/to current node and derive eccentricity
|
||
|
|
dist = dict(nx.single_source_shortest_path_length(G, current))
|
||
|
|
if len(dist) != N:
|
||
|
|
msg = "Cannot compute metric because graph is not connected."
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
current_ecc = max(dist.values())
|
||
|
|
|
||
|
|
# print status update
|
||
|
|
# print ("ecc of " + str(current) + " (" + str(ecc_lower[current]) + "/"
|
||
|
|
# + str(ecc_upper[current]) + ", deg: " + str(dist[current]) + ") is "
|
||
|
|
# + str(current_ecc))
|
||
|
|
# print(ecc_upper)
|
||
|
|
|
||
|
|
# (re)set bound extremes
|
||
|
|
maxuppernode = None
|
||
|
|
minlowernode = None
|
||
|
|
|
||
|
|
# update node bounds
|
||
|
|
for i in candidates:
|
||
|
|
# update eccentricity bounds
|
||
|
|
d = dist[i]
|
||
|
|
ecc_lower[i] = low = max(ecc_lower[i], max(d, (current_ecc - d)))
|
||
|
|
ecc_upper[i] = upp = min(ecc_upper[i], current_ecc + d)
|
||
|
|
|
||
|
|
# update min/max values of lower and upper bounds
|
||
|
|
minlower = min(ecc_lower[i], minlower)
|
||
|
|
maxlower = max(ecc_lower[i], maxlower)
|
||
|
|
minupper = min(ecc_upper[i], minupper)
|
||
|
|
maxupper = max(ecc_upper[i], maxupper)
|
||
|
|
|
||
|
|
# update candidate set
|
||
|
|
if compute == "diameter":
|
||
|
|
ruled_out = {
|
||
|
|
i
|
||
|
|
for i in candidates
|
||
|
|
if ecc_upper[i] <= maxlower and 2 * ecc_lower[i] >= maxupper
|
||
|
|
}
|
||
|
|
|
||
|
|
elif compute == "radius":
|
||
|
|
ruled_out = {
|
||
|
|
i
|
||
|
|
for i in candidates
|
||
|
|
if ecc_lower[i] >= minupper and ecc_upper[i] + 1 <= 2 * minlower
|
||
|
|
}
|
||
|
|
|
||
|
|
elif compute == "periphery":
|
||
|
|
ruled_out = {
|
||
|
|
i
|
||
|
|
for i in candidates
|
||
|
|
if ecc_upper[i] < maxlower
|
||
|
|
and (maxlower == maxupper or ecc_lower[i] > maxupper)
|
||
|
|
}
|
||
|
|
|
||
|
|
elif compute == "center":
|
||
|
|
ruled_out = {
|
||
|
|
i
|
||
|
|
for i in candidates
|
||
|
|
if ecc_lower[i] > minupper
|
||
|
|
and (minlower == minupper or ecc_upper[i] + 1 < 2 * minlower)
|
||
|
|
}
|
||
|
|
|
||
|
|
elif compute == "eccentricities":
|
||
|
|
ruled_out = {}
|
||
|
|
|
||
|
|
ruled_out.update(i for i in candidates if ecc_lower[i] == ecc_upper[i])
|
||
|
|
candidates -= ruled_out
|
||
|
|
|
||
|
|
# for i in ruled_out:
|
||
|
|
# print("removing %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"%
|
||
|
|
# (i,ecc_upper[i],maxlower,ecc_lower[i],maxupper))
|
||
|
|
# print("node %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"%
|
||
|
|
# (4,ecc_upper[4],maxlower,ecc_lower[4],maxupper))
|
||
|
|
# print("NODE 4: %g"%(ecc_upper[4] <= maxlower))
|
||
|
|
# print("NODE 4: %g"%(2 * ecc_lower[4] >= maxupper))
|
||
|
|
# print("NODE 4: %g"%(ecc_upper[4] <= maxlower
|
||
|
|
# and 2 * ecc_lower[4] >= maxupper))
|
||
|
|
|
||
|
|
# updating maxuppernode and minlowernode for selection in next round
|
||
|
|
for i in candidates:
|
||
|
|
if (
|
||
|
|
minlowernode is None
|
||
|
|
or (
|
||
|
|
ecc_lower[i] == ecc_lower[minlowernode]
|
||
|
|
and degrees[i] > degrees[minlowernode]
|
||
|
|
)
|
||
|
|
or (ecc_lower[i] < ecc_lower[minlowernode])
|
||
|
|
):
|
||
|
|
minlowernode = i
|
||
|
|
|
||
|
|
if (
|
||
|
|
maxuppernode is None
|
||
|
|
or (
|
||
|
|
ecc_upper[i] == ecc_upper[maxuppernode]
|
||
|
|
and degrees[i] > degrees[maxuppernode]
|
||
|
|
)
|
||
|
|
or (ecc_upper[i] > ecc_upper[maxuppernode])
|
||
|
|
):
|
||
|
|
maxuppernode = i
|
||
|
|
|
||
|
|
# print status update
|
||
|
|
# print (" min=" + str(minlower) + "/" + str(minupper) +
|
||
|
|
# " max=" + str(maxlower) + "/" + str(maxupper) +
|
||
|
|
# " candidates: " + str(len(candidates)))
|
||
|
|
# print("cand:",candidates)
|
||
|
|
# print("ecc_l",ecc_lower)
|
||
|
|
# print("ecc_u",ecc_upper)
|
||
|
|
# wait = input("press Enter to continue")
|
||
|
|
|
||
|
|
# return the correct value of the requested metric
|
||
|
|
if compute == "diameter":
|
||
|
|
return maxlower
|
||
|
|
elif compute == "radius":
|
||
|
|
return minupper
|
||
|
|
elif compute == "periphery":
|
||
|
|
p = [v for v in G if ecc_lower[v] == maxlower]
|
||
|
|
return p
|
||
|
|
elif compute == "center":
|
||
|
|
c = [v for v in G if ecc_upper[v] == minupper]
|
||
|
|
return c
|
||
|
|
elif compute == "eccentricities":
|
||
|
|
return ecc_lower
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
def eccentricity(G, v=None, sp=None):
|
||
|
|
"""Returns the eccentricity of nodes in G.
|
||
|
|
|
||
|
|
The eccentricity of a node v is the maximum distance from v to
|
||
|
|
all other nodes in G.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
v : node, optional
|
||
|
|
Return value of specified node
|
||
|
|
|
||
|
|
sp : dict of dicts, optional
|
||
|
|
All pairs shortest path lengths as a dictionary of dictionaries
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
ecc : dictionary
|
||
|
|
A dictionary of eccentricity values keyed by node.
|
||
|
|
"""
|
||
|
|
# if v is None: # none, use entire graph
|
||
|
|
# nodes=G.nodes()
|
||
|
|
# elif v in G: # is v a single node
|
||
|
|
# nodes=[v]
|
||
|
|
# else: # assume v is a container of nodes
|
||
|
|
# nodes=v
|
||
|
|
order = G.order()
|
||
|
|
|
||
|
|
e = {}
|
||
|
|
for n in G.nbunch_iter(v):
|
||
|
|
if sp is None:
|
||
|
|
length = nx.single_source_shortest_path_length(G, n)
|
||
|
|
L = len(length)
|
||
|
|
else:
|
||
|
|
try:
|
||
|
|
length = sp[n]
|
||
|
|
L = len(length)
|
||
|
|
except TypeError as e:
|
||
|
|
raise nx.NetworkXError('Format of "sp" is invalid.') from e
|
||
|
|
if L != order:
|
||
|
|
if G.is_directed():
|
||
|
|
msg = (
|
||
|
|
"Found infinite path length because the digraph is not"
|
||
|
|
" strongly connected"
|
||
|
|
)
|
||
|
|
else:
|
||
|
|
msg = "Found infinite path length because the graph is not" " connected"
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
|
||
|
|
e[n] = max(length.values())
|
||
|
|
|
||
|
|
if v in G:
|
||
|
|
return e[v] # return single value
|
||
|
|
else:
|
||
|
|
return e
|
||
|
|
|
||
|
|
|
||
|
|
def diameter(G, e=None, usebounds=False):
|
||
|
|
"""Returns the diameter of the graph G.
|
||
|
|
|
||
|
|
The diameter is the maximum eccentricity.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
e : eccentricity dictionary, optional
|
||
|
|
A precomputed dictionary of eccentricities.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
d : integer
|
||
|
|
Diameter of graph
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
eccentricity
|
||
|
|
"""
|
||
|
|
if usebounds is True and e is None and not G.is_directed():
|
||
|
|
return extrema_bounding(G, compute="diameter")
|
||
|
|
if e is None:
|
||
|
|
e = eccentricity(G)
|
||
|
|
return max(e.values())
|
||
|
|
|
||
|
|
|
||
|
|
def periphery(G, e=None, usebounds=False):
|
||
|
|
"""Returns the periphery of the graph G.
|
||
|
|
|
||
|
|
The periphery is the set of nodes with eccentricity equal to the diameter.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
e : eccentricity dictionary, optional
|
||
|
|
A precomputed dictionary of eccentricities.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
p : list
|
||
|
|
List of nodes in periphery
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
barycenter
|
||
|
|
center
|
||
|
|
"""
|
||
|
|
if usebounds is True and e is None and not G.is_directed():
|
||
|
|
return extrema_bounding(G, compute="periphery")
|
||
|
|
if e is None:
|
||
|
|
e = eccentricity(G)
|
||
|
|
diameter = max(e.values())
|
||
|
|
p = [v for v in e if e[v] == diameter]
|
||
|
|
return p
|
||
|
|
|
||
|
|
|
||
|
|
def radius(G, e=None, usebounds=False):
|
||
|
|
"""Returns the radius of the graph G.
|
||
|
|
|
||
|
|
The radius is the minimum eccentricity.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
e : eccentricity dictionary, optional
|
||
|
|
A precomputed dictionary of eccentricities.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
r : integer
|
||
|
|
Radius of graph
|
||
|
|
"""
|
||
|
|
if usebounds is True and e is None and not G.is_directed():
|
||
|
|
return extrema_bounding(G, compute="radius")
|
||
|
|
if e is None:
|
||
|
|
e = eccentricity(G)
|
||
|
|
return min(e.values())
|
||
|
|
|
||
|
|
|
||
|
|
def center(G, e=None, usebounds=False):
|
||
|
|
"""Returns the center of the graph G.
|
||
|
|
|
||
|
|
The center is the set of nodes with eccentricity equal to radius.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
e : eccentricity dictionary, optional
|
||
|
|
A precomputed dictionary of eccentricities.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
c : list
|
||
|
|
List of nodes in center
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
barycenter
|
||
|
|
periphery
|
||
|
|
"""
|
||
|
|
if usebounds is True and e is None and not G.is_directed():
|
||
|
|
return extrema_bounding(G, compute="center")
|
||
|
|
if e is None:
|
||
|
|
e = eccentricity(G)
|
||
|
|
radius = min(e.values())
|
||
|
|
p = [v for v in e if e[v] == radius]
|
||
|
|
return p
|
||
|
|
|
||
|
|
|
||
|
|
def barycenter(G, weight=None, attr=None, sp=None):
|
||
|
|
r"""Calculate barycenter of a connected graph, optionally with edge weights.
|
||
|
|
|
||
|
|
The :dfn:`barycenter` a
|
||
|
|
:func:`connected <networkx.algorithms.components.is_connected>` graph
|
||
|
|
:math:`G` is the subgraph induced by the set of its nodes :math:`v`
|
||
|
|
minimizing the objective function
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
\sum_{u \in V(G)} d_G(u, v),
|
||
|
|
|
||
|
|
where :math:`d_G` is the (possibly weighted) :func:`path length
|
||
|
|
<networkx.algorithms.shortest_paths.generic.shortest_path_length>`.
|
||
|
|
The barycenter is also called the :dfn:`median`. See [West01]_, p. 78.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : :class:`networkx.Graph`
|
||
|
|
The connected graph :math:`G`.
|
||
|
|
weight : :class:`str`, optional
|
||
|
|
Passed through to
|
||
|
|
:func:`~networkx.algorithms.shortest_paths.generic.shortest_path_length`.
|
||
|
|
attr : :class:`str`, optional
|
||
|
|
If given, write the value of the objective function to each node's
|
||
|
|
`attr` attribute. Otherwise do not store the value.
|
||
|
|
sp : dict of dicts, optional
|
||
|
|
All pairs shortest path lengths as a dictionary of dictionaries
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
list
|
||
|
|
Nodes of `G` that induce the barycenter of `G`.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
NetworkXNoPath
|
||
|
|
If `G` is disconnected. `G` may appear disconnected to
|
||
|
|
:func:`barycenter` if `sp` is given but is missing shortest path
|
||
|
|
lengths for any pairs.
|
||
|
|
ValueError
|
||
|
|
If `sp` and `weight` are both given.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
center
|
||
|
|
periphery
|
||
|
|
"""
|
||
|
|
if sp is None:
|
||
|
|
sp = nx.shortest_path_length(G, weight=weight)
|
||
|
|
else:
|
||
|
|
sp = sp.items()
|
||
|
|
if weight is not None:
|
||
|
|
raise ValueError("Cannot use both sp, weight arguments together")
|
||
|
|
smallest, barycenter_vertices, n = float("inf"), [], len(G)
|
||
|
|
for v, dists in sp:
|
||
|
|
if len(dists) < n:
|
||
|
|
raise nx.NetworkXNoPath(
|
||
|
|
f"Input graph {G} is disconnected, so every induced subgraph "
|
||
|
|
"has infinite barycentricity."
|
||
|
|
)
|
||
|
|
barycentricity = sum(dists.values())
|
||
|
|
if attr is not None:
|
||
|
|
G.nodes[v][attr] = barycentricity
|
||
|
|
if barycentricity < smallest:
|
||
|
|
smallest = barycentricity
|
||
|
|
barycenter_vertices = [v]
|
||
|
|
elif barycentricity == smallest:
|
||
|
|
barycenter_vertices.append(v)
|
||
|
|
return barycenter_vertices
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_submatrix(node, mat, node_list):
|
||
|
|
"""Removes row/col from a sparse matrix and returns the submatrix
|
||
|
|
"""
|
||
|
|
j = node_list.index(node)
|
||
|
|
n = list(range(len(node_list)))
|
||
|
|
n.pop(j)
|
||
|
|
|
||
|
|
if mat.shape[0] != mat.shape[1]:
|
||
|
|
raise nx.NetworkXError("Matrix must be square")
|
||
|
|
elif len(node_list) != mat.shape[0]:
|
||
|
|
msg = "Node list length does not match matrix dimentions"
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
|
||
|
|
mat = mat.tocsr()
|
||
|
|
mat = mat[n, :]
|
||
|
|
|
||
|
|
mat = mat.tocsc()
|
||
|
|
mat = mat[:, n]
|
||
|
|
|
||
|
|
node_list.pop(j)
|
||
|
|
|
||
|
|
return mat, node_list
|
||
|
|
|
||
|
|
|
||
|
|
def _count_lu_permutations(perm_array):
|
||
|
|
"""Counts the number of permutations in SuperLU perm_c or perm_r
|
||
|
|
"""
|
||
|
|
perm_cnt = 0
|
||
|
|
arr = perm_array.tolist()
|
||
|
|
for i in range(len(arr)):
|
||
|
|
if i != arr[i]:
|
||
|
|
perm_cnt += 1
|
||
|
|
n = arr.index(i)
|
||
|
|
arr[n] = arr[i]
|
||
|
|
arr[i] = i
|
||
|
|
|
||
|
|
return perm_cnt
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
def resistance_distance(G, nodeA, nodeB, weight=None, invert_weight=True):
|
||
|
|
"""Returns the resistance distance between node A and node B on graph G.
|
||
|
|
|
||
|
|
The resistance distance between two nodes of a graph is akin to treating
|
||
|
|
the graph as a grid of resistorses with a resistance equal to the provided
|
||
|
|
weight.
|
||
|
|
|
||
|
|
If weight is not provided, then a weight of 1 is used for all edges.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
A graph
|
||
|
|
|
||
|
|
nodeA : node
|
||
|
|
A node within graph G.
|
||
|
|
|
||
|
|
nodeB : node
|
||
|
|
A node within graph G, exclusive of Node A.
|
||
|
|
|
||
|
|
weight : string or None, optional (default=None)
|
||
|
|
The edge data key used to compute the resistance distance.
|
||
|
|
If None, then each edge has weight 1.
|
||
|
|
|
||
|
|
invert_weight : boolean (default=True)
|
||
|
|
Proper calculation of resistance distance requires building the
|
||
|
|
Laplacian matrix with the reciprocal of the weight. Not required
|
||
|
|
if the weight is already inverted. Weight cannot be zero.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rd : float
|
||
|
|
Value of effective resistance distance
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Overview discussion:
|
||
|
|
* https://en.wikipedia.org/wiki/Resistance_distance
|
||
|
|
* http://mathworld.wolfram.com/ResistanceDistance.html
|
||
|
|
|
||
|
|
Additional details:
|
||
|
|
Vaya Sapobi Samui Vos, “Methods for determining the effective resistance,” M.S.,
|
||
|
|
Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands, 2016
|
||
|
|
Available: `Link to thesis <https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/vos_vaya_master.pdf>`_
|
||
|
|
"""
|
||
|
|
import numpy as np
|
||
|
|
import scipy.sparse
|
||
|
|
|
||
|
|
if not nx.is_connected(G):
|
||
|
|
msg = "Graph G must be strongly connected."
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
elif nodeA not in G:
|
||
|
|
msg = "Node A is not in graph G."
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
elif nodeB not in G:
|
||
|
|
msg = "Node B is not in graph G."
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
elif nodeA == nodeB:
|
||
|
|
msg = "Node A and Node B cannot be the same."
|
||
|
|
raise nx.NetworkXError(msg)
|
||
|
|
|
||
|
|
G = G.copy()
|
||
|
|
node_list = list(G)
|
||
|
|
|
||
|
|
if invert_weight and weight is not None:
|
||
|
|
if G.is_multigraph():
|
||
|
|
for (u, v, k, d) in G.edges(keys=True, data=True):
|
||
|
|
d[weight] = 1 / d[weight]
|
||
|
|
else:
|
||
|
|
for (u, v, d) in G.edges(data=True):
|
||
|
|
d[weight] = 1 / d[weight]
|
||
|
|
# Replace with collapsing topology or approximated zero?
|
||
|
|
|
||
|
|
# Using determinants to compute the effective resistance is more memory
|
||
|
|
# efficent than directly calculating the psuedo-inverse
|
||
|
|
L = nx.laplacian_matrix(G, node_list, weight=weight)
|
||
|
|
|
||
|
|
Lsub_a, node_list_a = _laplacian_submatrix(nodeA, L.copy(), node_list[:])
|
||
|
|
Lsub_ab, node_list_ab = _laplacian_submatrix(nodeB, Lsub_a.copy(), node_list_a[:])
|
||
|
|
|
||
|
|
# Factorize Laplacian submatrixes and extract diagonals
|
||
|
|
# Order the diagonals to minimize the likelihood over overflows
|
||
|
|
# during computing the determinant
|
||
|
|
lu_a = scipy.sparse.linalg.splu(Lsub_a, options=dict(SymmetricMode=True))
|
||
|
|
LdiagA = lu_a.U.diagonal()
|
||
|
|
LdiagA_s = np.product(np.sign(LdiagA)) * np.product(lu_a.L.diagonal())
|
||
|
|
LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_r)
|
||
|
|
LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_c)
|
||
|
|
LdiagA = np.absolute(LdiagA)
|
||
|
|
LdiagA = np.sort(LdiagA)
|
||
|
|
|
||
|
|
lu_ab = scipy.sparse.linalg.splu(Lsub_ab, options=dict(SymmetricMode=True))
|
||
|
|
LdiagAB = lu_ab.U.diagonal()
|
||
|
|
LdiagAB_s = np.product(np.sign(LdiagAB)) * np.product(lu_ab.L.diagonal())
|
||
|
|
LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_r)
|
||
|
|
LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_c)
|
||
|
|
LdiagAB = np.absolute(LdiagAB)
|
||
|
|
LdiagAB = np.sort(LdiagAB)
|
||
|
|
|
||
|
|
# Calculate the ratio of determinant, rd = det(Lsub_ab)/det(Lsub_a)
|
||
|
|
Ldet = np.product(np.divide(np.append(LdiagAB, [1]), LdiagA))
|
||
|
|
rd = Ldet * LdiagAB_s / LdiagA_s
|
||
|
|
|
||
|
|
return rd
|