386 lines
13 KiB
Python
386 lines
13 KiB
Python
|
"""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, py_random_state
|
||
|
|
|
||
|
|
__all__ = ["random_reference", "lattice_reference", "sigma", "omega"]
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
|
||
|
|
|
||
|
|
@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
|