"""Functions for estimating the small-world-ness of graphs.
A small world network is characterized by a small average shortest path length,
and a large clustering coefficient.
Small-worldness is commonly measured with the coefficient sigma or omega.
Both coefficients compare the average clustering coefficient and shortest path
length of a given graph against the same quantities for an equivalent random
or lattice graph.
For more information, see the Wikipedia article on small-world network [1]_.
.. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network
"""
import networkx as nx
from networkx.utils import not_implemented_for
from networkx.utils import py_random_state
__all__ = ["random_reference", "lattice_reference", "sigma", "omega"]
[docs]@py_random_state(3)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def random_reference(G, niter=1, connectivity=True, seed=None):
"""Compute a random graph by swapping edges of a given graph.
Parameters
----------
G : graph
An undirected graph with 4 or more nodes.
niter : integer (optional, default=1)
An edge is rewired approximately `niter` times.
connectivity : boolean (optional, default=True)
When True, ensure connectivity for the randomized graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
G : graph
The randomized graph.
Notes
-----
The implementation is adapted from the algorithm by Maslov and Sneppen
(2002) [1]_.
References
----------
.. [1] Maslov, Sergei, and Kim Sneppen.
"Specificity and stability in topology of protein networks."
Science 296.5569 (2002): 910-913.
"""
if len(G) < 4:
raise nx.NetworkXError("Graph has less than four nodes.")
from networkx.utils import cumulative_distribution, discrete_sequence
local_conn = nx.connectivity.local_edge_connectivity
G = G.copy()
keys, degrees = zip(*G.degree()) # keys, degree
cdf = cumulative_distribution(degrees) # cdf of degree
nnodes = len(G)
nedges = nx.number_of_edges(G)
niter = niter * nedges
ntries = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2))
swapcount = 0
for i in range(niter):
n = 0
while n < ntries:
# pick two random edges without creating edge list
# choose source node indices from discrete distribution
(ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed)
if ai == ci:
continue # same source, skip
a = keys[ai] # convert index to label
c = keys[ci]
# choose target uniformly from neighbors
b = seed.choice(list(G.neighbors(a)))
d = seed.choice(list(G.neighbors(c)))
if b in [a, c, d] or d in [a, b, c]:
continue # all vertices should be different
# don't create parallel edges
if (d not in G[a]) and (b not in G[c]):
G.add_edge(a, d)
G.add_edge(c, b)
G.remove_edge(a, b)
G.remove_edge(c, d)
# Check if the graph is still connected
if connectivity and local_conn(G, a, b) == 0:
# Not connected, revert the swap
G.remove_edge(a, d)
G.remove_edge(c, b)
G.add_edge(a, b)
G.add_edge(c, d)
else:
swapcount += 1
break
n += 1
return G
[docs]@py_random_state(4)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def lattice_reference(G, niter=5, D=None, connectivity=True, seed=None):
"""Latticize the given graph by swapping edges.
Parameters
----------
G : graph
An undirected graph with 4 or more nodes.
niter : integer (optional, default=1)
An edge is rewired approximatively niter times.
D : numpy.array (optional, default=None)
Distance to the diagonal matrix.
connectivity : boolean (optional, default=True)
Ensure connectivity for the latticized graph when set to True.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
G : graph
The latticized graph.
Notes
-----
The implementation is adapted from the algorithm by Sporns et al. [1]_.
which is inspired from the original work by Maslov and Sneppen(2002) [2]_.
References
----------
.. [1] Sporns, Olaf, and Jonathan D. Zwi.
"The small world of the cerebral cortex."
Neuroinformatics 2.2 (2004): 145-162.
.. [2] Maslov, Sergei, and Kim Sneppen.
"Specificity and stability in topology of protein networks."
Science 296.5569 (2002): 910-913.
"""
import numpy as np
from networkx.utils import cumulative_distribution, discrete_sequence
local_conn = nx.connectivity.local_edge_connectivity
if len(G) < 4:
raise nx.NetworkXError("Graph has less than four nodes.")
# Instead of choosing uniformly at random from a generated edge list,
# this algorithm chooses nonuniformly from the set of nodes with
# probability weighted by degree.
G = G.copy()
keys, degrees = zip(*G.degree()) # keys, degree
cdf = cumulative_distribution(degrees) # cdf of degree
nnodes = len(G)
nedges = nx.number_of_edges(G)
if D is None:
D = np.zeros((nnodes, nnodes))
un = np.arange(1, nnodes)
um = np.arange(nnodes - 1, 0, -1)
u = np.append((0,), np.where(un < um, un, um))
for v in range(int(np.ceil(nnodes / 2))):
D[nnodes - v - 1, :] = np.append(u[v + 1 :], u[: v + 1])
D[v, :] = D[nnodes - v - 1, :][::-1]
niter = niter * nedges
# maximal number of rewiring attempts per 'niter'
max_attempts = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2))
for _ in range(niter):
n = 0
while n < max_attempts:
# pick two random edges without creating edge list
# choose source node indices from discrete distribution
(ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed)
if ai == ci:
continue # same source, skip
a = keys[ai] # convert index to label
c = keys[ci]
# choose target uniformly from neighbors
b = seed.choice(list(G.neighbors(a)))
d = seed.choice(list(G.neighbors(c)))
bi = keys.index(b)
di = keys.index(d)
if b in [a, c, d] or d in [a, b, c]:
continue # all vertices should be different
# don't create parallel edges
if (d not in G[a]) and (b not in G[c]):
if D[ai, bi] + D[ci, di] >= D[ai, ci] + D[bi, di]:
# only swap if we get closer to the diagonal
G.add_edge(a, d)
G.add_edge(c, b)
G.remove_edge(a, b)
G.remove_edge(c, d)
# Check if the graph is still connected
if connectivity and local_conn(G, a, b) == 0:
# Not connected, revert the swap
G.remove_edge(a, d)
G.remove_edge(c, b)
G.add_edge(a, b)
G.add_edge(c, d)
else:
break
n += 1
return G
[docs]@py_random_state(3)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def sigma(G, niter=100, nrand=10, seed=None):
"""Returns the small-world coefficient (sigma) of the given graph.
The small-world coefficient is defined as:
sigma = C/Cr / L/Lr
where C and L are respectively the average clustering coefficient and
average shortest path length of G. Cr and Lr are respectively the average
clustering coefficient and average shortest path length of an equivalent
random graph.
A graph is commonly classified as small-world if sigma>1.
Parameters
----------
G : NetworkX graph
An undirected graph.
niter : integer (optional, default=100)
Approximate number of rewiring per edge to compute the equivalent
random graph.
nrand : integer (optional, default=10)
Number of random graphs generated to compute the average clustering
coefficient (Cr) and average shortest path length (Lr).
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
sigma : float
The small-world coefficient of G.
Notes
-----
The implementation is adapted from Humphries et al. [1]_ [2]_.
References
----------
.. [1] The brainstem reticular formation is a small-world, not scale-free,
network M. D. Humphries, K. Gurney and T. J. Prescott,
Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354.
.. [2] Humphries and Gurney (2008).
"Network 'Small-World-Ness': A Quantitative Method for Determining
Canonical Network Equivalence".
PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051.
"""
import numpy as np
# Compute the mean clustering coefficient and average shortest path length
# for an equivalent random graph
randMetrics = {"C": [], "L": []}
for i in range(nrand):
Gr = random_reference(G, niter=niter, seed=seed)
randMetrics["C"].append(nx.transitivity(Gr))
randMetrics["L"].append(nx.average_shortest_path_length(Gr))
C = nx.transitivity(G)
L = nx.average_shortest_path_length(G)
Cr = np.mean(randMetrics["C"])
Lr = np.mean(randMetrics["L"])
sigma = (C / Cr) / (L / Lr)
return sigma
[docs]@py_random_state(3)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def omega(G, niter=5, nrand=10, seed=None):
"""Returns the small-world coefficient (omega) of a graph
The small-world coefficient of a graph G is:
omega = Lr/L - C/Cl
where C and L are respectively the average clustering coefficient and
average shortest path length of G. Lr is the average shortest path length
of an equivalent random graph and Cl is the average clustering coefficient
of an equivalent lattice graph.
The small-world coefficient (omega) measures how much G is like a lattice
or a random graph. Negative values mean G is similar to a lattice whereas
positive values mean G is a random graph.
Values close to 0 mean that G has small-world characteristics.
Parameters
----------
G : NetworkX graph
An undirected graph.
niter: integer (optional, default=5)
Approximate number of rewiring per edge to compute the equivalent
random graph.
nrand: integer (optional, default=10)
Number of random graphs generated to compute the maximal clustering
coefficient (Cr) and average shortest path length (Lr).
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
omega : float
The small-world coefficient (omega)
Notes
-----
The implementation is adapted from the algorithm by Telesford et al. [1]_.
References
----------
.. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011).
"The Ubiquity of Small-World Networks".
Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451.
doi:10.1089/brain.2011.0038.
"""
import numpy as np
# Compute the mean clustering coefficient and average shortest path length
# for an equivalent random graph
randMetrics = {"C": [], "L": []}
# Calculate initial average clustering coefficient which potentially will
# get replaced by higher clustering coefficients from generated lattice
# reference graphs
Cl = nx.average_clustering(G)
niter_lattice_reference = niter
niter_random_reference = niter * 2
for _ in range(nrand):
# Generate random graph
Gr = random_reference(G, niter=niter_random_reference, seed=seed)
randMetrics["L"].append(nx.average_shortest_path_length(Gr))
# Generate lattice graph
Gl = lattice_reference(G, niter=niter_lattice_reference, seed=seed)
# Replace old clustering coefficient, if clustering is higher in
# generated lattice reference
Cl_temp = nx.average_clustering(Gl)
if Cl_temp > Cl:
Cl = Cl_temp
C = nx.average_clustering(G)
L = nx.average_shortest_path_length(G)
Lr = np.mean(randMetrics["L"])
omega = (Lr / L) - (C / Cl)
return omega