# -*- coding: utf-8 -*-
# Copyright (C) 2004-2018 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
# Authors: Aric Hagberg (hagberg@lanl.gov)
# Dan Schult (dschult@colgate.edu)
"""Graph diameter, radius, eccentricity and other properties."""
import networkx
__all__ = ['extrema_bounding', 'eccentricity', 'diameter',
'radius', 'periphery', 'center']
[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(networkx.single_source_shortest_path_length(G, current))
if len(dist) != N:
msg = ('Cannot compute metric because graph is not connected.')
raise networkx.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):
"""Return 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 = networkx.single_source_shortest_path_length(G, n)
L = len(length)
else:
try:
length = sp[n]
L = len(length)
except TypeError:
raise networkx.NetworkXError('Format of "sp" is invalid.')
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 networkx.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):
"""Return 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):
"""Return 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
"""
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):
"""Return 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):
"""Return 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
"""
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