"""Graph diameter, radius, eccentricity and other properties."""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"eccentricity",
"diameter",
"radius",
"periphery",
"center",
"barycenter",
"resistance_distance",
]
def _extrema_bounding(G, compute="diameter", weight=None):
"""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,
"eccentricities" for the maximum distance from each node to all other nodes in G
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
value : value of the requested metric
int for "diameter" and "radius" or
list of nodes for "center" and "periphery" or
dictionary of eccentricity values keyed by node for "eccentricities"
Raises
------
NetworkXError
If the graph consists of multiple components
ValueError
If `compute` is not one of "diameter", "radius", "periphery", "center", or "eccentricities".
Notes
-----
This algorithm was proposed in [1]_ and discussed further in [2]_ and [3]_.
References
----------
.. [1] F. W. Takes, W. A. Kosters,
"Determining the diameter of small world networks."
Proceedings of the 20th ACM international conference on Information and knowledge management, 2011
https://dl.acm.org/doi/abs/10.1145/2063576.2063748
.. [2] F. W. Takes, W. A. Kosters,
"Computing the Eccentricity Distribution of Large Graphs."
Algorithms, 2013
https://www.mdpi.com/1999-4893/6/1/100
.. [3] M. Borassi, P. Crescenzi, M. Habib, W. A. Kosters, A. Marino, F. W. Takes,
"Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs: With an application to the six degrees of separation games. "
Theoretical Computer Science, 2015
https://www.sciencedirect.com/science/article/pii/S0304397515001644
"""
# 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 = nx.shortest_path_length(G, source=current, weight=weight)
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 = set()
else:
msg = "compute must be one of 'diameter', 'radius', 'periphery', 'center', 'eccentricities'"
raise ValueError(msg)
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
if compute == "radius":
return minupper
if compute == "periphery":
p = [v for v in G if ecc_lower[v] == maxlower]
return p
if compute == "center":
c = [v for v in G if ecc_upper[v] == minupper]
return c
if compute == "eccentricities":
return ecc_lower
return None
[docs]def eccentricity(G, v=None, sp=None, weight=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
weight : string, function, or None (default=None)
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
ecc : dictionary
A dictionary of eccentricity values keyed by node.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> dict(nx.eccentricity(G))
{1: 2, 2: 3, 3: 2, 4: 2, 5: 3}
>>> dict(nx.eccentricity(G, v=[1, 5])) # This returns the eccentrity of node 1 & 5
{1: 2, 5: 3}
"""
# 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.shortest_path_length(G, source=n, weight=weight)
L = len(length)
else:
try:
length = sp[n]
L = len(length)
except TypeError as err:
raise nx.NetworkXError('Format of "sp" is invalid.') from err
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
return e
[docs]def diameter(G, e=None, usebounds=False, weight=None):
"""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.
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
d : integer
Diameter of graph
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> nx.diameter(G)
3
See Also
--------
eccentricity
"""
if usebounds is True and e is None and not G.is_directed():
return _extrema_bounding(G, compute="diameter", weight=weight)
if e is None:
e = eccentricity(G, weight=weight)
return max(e.values())
[docs]def periphery(G, e=None, usebounds=False, weight=None):
"""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.
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
p : list
List of nodes in periphery
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> nx.periphery(G)
[2, 5]
See Also
--------
barycenter
center
"""
if usebounds is True and e is None and not G.is_directed():
return _extrema_bounding(G, compute="periphery", weight=weight)
if e is None:
e = eccentricity(G, weight=weight)
diameter = max(e.values())
p = [v for v in e if e[v] == diameter]
return p
[docs]def radius(G, e=None, usebounds=False, weight=None):
"""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.
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
r : integer
Radius of graph
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> nx.radius(G)
2
"""
if usebounds is True and e is None and not G.is_directed():
return _extrema_bounding(G, compute="radius", weight=weight)
if e is None:
e = eccentricity(G, weight=weight)
return min(e.values())
[docs]def center(G, e=None, usebounds=False, weight=None):
"""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.
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
c : list
List of nodes in center
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> list(nx.center(G))
[1, 3, 4]
See Also
--------
barycenter
periphery
"""
if usebounds is True and e is None and not G.is_directed():
return _extrema_bounding(G, compute="center", weight=weight)
if e is None:
e = eccentricity(G, weight=weight)
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.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> nx.barycenter(G)
[1, 3, 4]
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 _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
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> nx.resistance_distance(G, 1, 3)
0.625
Notes
-----
Overviews are provided in [1]_ and [2]_. Additional details on computational
methods, proofs of properties, and corresponding MATLAB codes are provided
in [3]_.
References
----------
.. [1] Wikipedia
"Resistance distance."
https://en.wikipedia.org/wiki/Resistance_distance
.. [2] E. W. Weisstein
"Resistance Distance."
MathWorld--A Wolfram Web Resource
https://mathworld.wolfram.com/ResistanceDistance.html
.. [3] V. S. S. Vos,
"Methods for determining the effective resistance."
Mestrado, Mathematisch Instituut Universiteit Leiden, 2016
https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/vos_vaya_master.pdf
"""
import numpy as np
import scipy as sp
import scipy.sparse.linalg # call as sp.sparse.linalg
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
# efficient than directly calculating the pseudo-inverse
L = nx.laplacian_matrix(G, node_list, weight=weight).asformat("csc")
indices = list(range(L.shape[0]))
# w/ nodeA removed
indices.remove(node_list.index(nodeA))
L_a = L[indices, :][:, indices]
# Both nodeA and nodeB removed
indices.remove(node_list.index(nodeB))
L_ab = L[indices, :][:, indices]
# Factorize Laplacian submatrixes and extract diagonals
# Order the diagonals to minimize the likelihood over overflows
# during computing the determinant
lu_a = sp.sparse.linalg.splu(L_a, options={"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 = sp.sparse.linalg.splu(L_ab, options={"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(L_ab)/det(L_a)
Ldet = np.product(np.divide(np.append(LdiagAB, [1]), LdiagA))
rd = Ldet * LdiagAB_s / LdiagA_s
return rd