# Source code for networkx.algorithms.distance_measures

```
"""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",
"kemeny_constant",
"effective_graph_resistance",
]
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]
@nx._dispatchable(edge_attrs="weight")
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 eccentricity 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]
@nx._dispatchable(edge_attrs="weight")
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]
@nx._dispatchable(edge_attrs="weight")
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]
@nx._dispatchable(edge_attrs="weight")
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]
@nx._dispatchable(edge_attrs="weight")
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]
@nx._dispatchable(edge_attrs="weight", mutates_input={"attr": 2})
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)
if attr is not None:
nx._clear_cache(G)
return barycenter_vertices
[docs]
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def resistance_distance(G, nodeA=None, nodeB=None, weight=None, invert_weight=True):
"""Returns the resistance distance between pairs of nodes in graph G.
The resistance distance between two nodes of a graph is akin to treating
the graph as a grid of resistors with a resistance equal to the provided
weight [1]_, [2]_.
If weight is not provided, then a weight of 1 is used for all edges.
If two nodes are the same, the resistance distance is zero.
Parameters
----------
G : NetworkX graph
A graph
nodeA : node or None, optional (default=None)
A node within graph G.
If None, compute resistance distance using all nodes as source nodes.
nodeB : node or None, optional (default=None)
A node within graph G.
If None, compute resistance distance using all nodes as target nodes.
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 : dict or float
If `nodeA` and `nodeB` are given, resistance distance between `nodeA`
and `nodeB`. If `nodeA` or `nodeB` is unspecified (the default), a
dictionary of nodes with resistance distances as the value.
Raises
------
NetworkXNotImplemented
If `G` is a directed graph.
NetworkXError
If `G` is not connected, or contains no nodes,
or `nodeA` is not in `G` or `nodeB` is not in `G`.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> round(nx.resistance_distance(G, 1, 3), 10)
0.625
Notes
-----
The implementation is based on Theorem A in [2]_. Self-loops are ignored.
Multi-edges are contracted in one edge with weight equal to the harmonic sum of the weights.
References
----------
.. [1] Wikipedia
"Resistance distance."
https://en.wikipedia.org/wiki/Resistance_distance
.. [2] D. J. Klein and M. Randic.
Resistance distance.
J. of Math. Chem. 12:81-95, 1993.
"""
import numpy as np
if len(G) == 0:
raise nx.NetworkXError("Graph G must contain at least one node.")
if not nx.is_connected(G):
raise nx.NetworkXError("Graph G must be strongly connected.")
if nodeA is not None and nodeA not in G:
raise nx.NetworkXError("Node A is not in graph G.")
if nodeB is not None and nodeB not in G:
raise nx.NetworkXError("Node B is not in graph G.")
G = G.copy()
node_list = list(G)
# Invert weights
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]
# Compute resistance distance using the Pseudo-inverse of the Laplacian
# Self-loops are ignored
L = nx.laplacian_matrix(G, weight=weight).todense()
Linv = np.linalg.pinv(L, hermitian=True)
# Return relevant distances
if nodeA is not None and nodeB is not None:
i = node_list.index(nodeA)
j = node_list.index(nodeB)
return Linv.item(i, i) + Linv.item(j, j) - Linv.item(i, j) - Linv.item(j, i)
elif nodeA is not None:
i = node_list.index(nodeA)
d = {}
for n in G:
j = node_list.index(n)
d[n] = Linv.item(i, i) + Linv.item(j, j) - Linv.item(i, j) - Linv.item(j, i)
return d
elif nodeB is not None:
j = node_list.index(nodeB)
d = {}
for n in G:
i = node_list.index(n)
d[n] = Linv.item(i, i) + Linv.item(j, j) - Linv.item(i, j) - Linv.item(j, i)
return d
else:
d = {}
for n in G:
i = node_list.index(n)
d[n] = {}
for n2 in G:
j = node_list.index(n2)
d[n][n2] = (
Linv.item(i, i)
+ Linv.item(j, j)
- Linv.item(i, j)
- Linv.item(j, i)
)
return d
[docs]
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def effective_graph_resistance(G, weight=None, invert_weight=True):
"""Returns the Effective graph resistance of G.
Also known as the Kirchhoff index.
The effective graph resistance is defined as the sum
of the resistance distance of every node pair in G [1]_.
If weight is not provided, then a weight of 1 is used for all edges.
The effective graph resistance of a disconnected graph is infinite.
Parameters
----------
G : NetworkX graph
A graph
weight : string or None, optional (default=None)
The edge data key used to compute the effective graph resistance.
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
-------
RG : float
The effective graph resistance of `G`.
Raises
------
NetworkXNotImplemented
If `G` is a directed graph.
NetworkXError
If `G` does not contain any nodes.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> round(nx.effective_graph_resistance(G), 10)
10.25
Notes
-----
The implementation is based on Theorem 2.2 in [2]_. Self-loops are ignored.
Multi-edges are contracted in one edge with weight equal to the harmonic sum of the weights.
References
----------
.. [1] Wolfram
"Kirchhoff Index."
https://mathworld.wolfram.com/KirchhoffIndex.html
.. [2] W. Ellens, F. M. Spieksma, P. Van Mieghem, A. Jamakovic, R. E. Kooij.
Effective graph resistance.
Lin. Alg. Appl. 435:2491-2506, 2011.
"""
import numpy as np
if len(G) == 0:
raise nx.NetworkXError("Graph G must contain at least one node.")
# Disconnected graphs have infinite Effective graph resistance
if not nx.is_connected(G):
return float("inf")
# Invert weights
G = G.copy()
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]
# Get Laplacian eigenvalues
mu = np.sort(nx.laplacian_spectrum(G, weight=weight))
# Compute Effective graph resistance based on spectrum of the Laplacian
# Self-loops are ignored
return float(np.sum(1 / mu[1:]) * G.number_of_nodes())
[docs]
@nx.utils.not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def kemeny_constant(G, *, weight=None):
"""Returns the Kemeny constant of the given graph.
The *Kemeny constant* (or Kemeny's constant) of a graph `G`
can be computed by regarding the graph as a Markov chain.
The Kemeny constant is then the expected number of time steps
to transition from a starting state i to a random destination state
sampled from the Markov chain's stationary distribution.
The Kemeny constant is independent of the chosen initial state [1]_.
The Kemeny constant measures the time needed for spreading
across a graph. Low values indicate a closely connected graph
whereas high values indicate a spread-out graph.
If weight is not provided, then a weight of 1 is used for all edges.
Since `G` represents a Markov chain, the weights must be positive.
Parameters
----------
G : NetworkX graph
weight : string or None, optional (default=None)
The edge data key used to compute the Kemeny constant.
If None, then each edge has weight 1.
Returns
-------
float
The Kemeny constant of the graph `G`.
Raises
------
NetworkXNotImplemented
If the graph `G` is directed.
NetworkXError
If the graph `G` is not connected, or contains no nodes,
or has edges with negative weights.
Examples
--------
>>> G = nx.complete_graph(5)
>>> round(nx.kemeny_constant(G), 10)
3.2
Notes
-----
The implementation is based on equation (3.3) in [2]_.
Self-loops are allowed and indicate a Markov chain where
the state can remain the same. Multi-edges are contracted
in one edge with weight equal to the sum of the weights.
References
----------
.. [1] Wikipedia
"Kemeny's constant."
https://en.wikipedia.org/wiki/Kemeny%27s_constant
.. [2] Lovász L.
Random walks on graphs: A survey.
Paul Erdös is Eighty, vol. 2, Bolyai Society,
Mathematical Studies, Keszthely, Hungary (1993), pp. 1-46
"""
import numpy as np
import scipy as sp
if len(G) == 0:
raise nx.NetworkXError("Graph G must contain at least one node.")
if not nx.is_connected(G):
raise nx.NetworkXError("Graph G must be connected.")
if nx.is_negatively_weighted(G, weight=weight):
raise nx.NetworkXError("The weights of graph G must be nonnegative.")
# Compute matrix H = D^-1/2 A D^-1/2
A = nx.adjacency_matrix(G, weight=weight)
n, m = A.shape
diags = A.sum(axis=1)
with np.errstate(divide="ignore"):
diags_sqrt = 1.0 / np.sqrt(diags)
diags_sqrt[np.isinf(diags_sqrt)] = 0
DH = sp.sparse.csr_array(sp.sparse.spdiags(diags_sqrt, 0, m, n, format="csr"))
H = DH @ (A @ DH)
# Compute eigenvalues of H
eig = np.sort(sp.linalg.eigvalsh(H.todense()))
# Compute the Kemeny constant
return float(np.sum(1 / (1 - eig[:-1])))
```