"""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",
]
[docs]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
[docs]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
[docs]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())
[docs]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
[docs]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())
[docs]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
[docs]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
[docs]@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