def omega_sigma(matrix):
"""Returns the small-world coefficients (omega & sigma) of a graph.
Omega ranges between -1 and 1. Values close to 0 mean the matrix
features small-world characteristics.
Values close to -1 mean the network has a lattice structure and values
close to 1 mean G is a random network.
A network is commonly classified as small-world if sigma > 1.
Parameters
----------
matrix : numpy.ndarray
A weighted undirected graph.
Returns
-------
smallworld : tuple of float
The small-work coefficients (omega & sigma).
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.
"""
transitivity_rand_list = []
transitivity_latt_list = []
path_length_rand_list = []
for i in range(10):
logging.debug('Generating random and lattice matrices, '
'iteration #{}.'.format(i))
random = bct.randmio_und(matrix, 10)[0]
lattice = bct.latmio_und(matrix, 10)[1]
transitivity_rand_list.append(bct.transitivity_wu(random))
transitivity_latt_list.append(bct.transitivity_wu(lattice))
path_length_rand_list.append(avg_cast(bct.distance_wei(random)[0]))
transitivity = bct.transitivity_wu(matrix)
path_length = avg_cast(bct.distance_wei(matrix)[0])
transitivity_rand = np.mean(transitivity_rand_list)
transitivity_latt = np.mean(transitivity_latt_list)
path_length_rand = np.mean(path_length_rand_list)
omega = (path_length_rand / path_length) - \
(transitivity / transitivity_latt)
sigma = (transitivity / transitivity_rand) / \
(path_length / path_length_rand)
return float(omega), float(sigma)
def get_true_network_metrics(A):
#control centrality
c_c = control_centrality(A)
cc_fake = np.zeros((100, 1))
for i in range(0, 100):
cc_fake[i] = np.mean(control_centrality(generate_fake_graph(A)))
m_cc_fake = np.mean(cc_fake)
cc_norm = c_c / m_cc_fake
# Get identity of node with lowest control centrality
min_cc_true = np.where(c_c == np.amin(c_c))[0]
# get synchronizability
sync = synchronizability(A)
# normalized sync
sync_fake = np.zeros((100, 1))
for i in range(0, 100):
sync_fake[i] = synchronizability(generate_fake_graph(A))
m_sync_fake = np.mean(sync_fake)
sync_norm = sync / m_sync_fake
# get betweeness centrality
bc = betweenness_centrality(A)
bc_fake = np.zeros((100, 1))
for i in range(0, 100):
bc_fake[i] = np.mean(betweenness_centrality(generate_fake_graph(A)))
m_bc_fake = np.mean(bc_fake)
bc_norm = bc / m_bc_fake
# Get identity of node with max bc
max_bc_true = np.where(bc == np.amax(bc))[0]
# get eigenvector centrality
ec = bct.eigenvector_centrality_und(A)
ec_fake = np.zeros((100, 1))
for i in range(0, 100):
ec_fake[i] = np.mean(
bct.eigenvector_centrality_und(generate_fake_graph(A)))
m_ec_fake = np.mean(ec_fake)
ec_norm = ec / m_ec_fake
# Get identity of node with max ec
max_ec_true = np.where(ec == np.amax(ec))[0]
# get edge betweeness centrality
edge_bc, ignore = bct.edge_betweenness_wei(A)
edge_bc_fake = np.zeros((100, 1))
for i in range(0, 100):
edge_bc_fake[i] = np.mean(
bct.edge_betweenness_wei(generate_fake_graph(A))[0])
m_edge_bc_fake = np.mean(edge_bc_fake)
edge_bc_norm = edge_bc / m_edge_bc_fake
# get clustering coeff
clust = bct.clustering_coef_wu(A)
clust_fake = np.zeros((100, 1))
for i in range(0, 100):
clust_fake[i] = np.mean(bct.clustering_coef_wu(generate_fake_graph(A)))
m_clust_fake = np.mean(clust_fake)
clust_norm = clust / m_clust_fake
# Get identity of node with max clust
max_clust_true = np.where(clust == np.amax(clust))[0]
# get node strength
ns = node_strength(A)
ns_fake = np.zeros((100, 1))
for i in range(0, 100):
ns_fake[i] = np.mean(node_strength(generate_fake_graph(A)))
m_ns_fake = np.mean(ns_fake)
ns_norm = ns / m_ns_fake
# Get identity of node with max clust
max_ns_true = np.where(ns == np.amax(ns))[0]
#Get true efficiency
Ci, ignore = bct.modularity_und(A)
par = bct.participation_coef(A, Ci)
eff = bct.efficiency_wei(A, 0)
eff_fake = np.zeros((100, 1))
for i in range(0, 100):
eff_fake[i] = (bct.efficiency_wei(generate_fake_graph(A)))
m_eff_fake = np.mean(eff_fake)
eff_norm = eff / m_eff_fake
# Get true transistivity
trans = bct.transitivity_wu(A)
trans_fake = np.zeros((100, 1))
for i in range(0, 100):
trans_fake[i] = (bct.transitivity_wu(generate_fake_graph(A)))
m_trans_fake = np.mean(trans_fake)
trans_norm = trans / m_trans_fake
# store output results in a dictionary
#nodal
results = {}
results['control_centrality'] = c_c
results['control_centrality_norm'] = cc_norm
results['min_cc_node'] = min_cc_true
# global measure
results['sync'] = sync
results['sync_norm'] = sync_norm
# nodal
results['bc'] = bc
results['bc_norm'] = bc_norm
results['max_bc_node'] = max_bc_true
# nodal
results['ec'] = ec
results['ec_norm'] = ec_norm
results['max_ec_node'] = max_ec_true
# nodal
results['clust'] = clust
results['clust_norm'] = clust_norm
results['max_clust_node'] = max_clust_true
# nodal
results['ns'] = ns
results['ns_norm'] = ns_norm
results['max_ns_node'] = max_ns_true
# global
results['eff'] = eff
results['eff_norm'] = eff_norm
# global
results['trans'] = trans
results['trans_norm'] = trans_norm
# nodal
results['par'] = par
# edge
results['edge_bc'] = edge_bc
results['edge_bc_norm'] = edge_bc_norm
return (results)
def test_transitivity_wu(): x = load_sample(thres=.23) t = bct.transitivity_wu(x) assert np.allclose(t, 1.32927829)
def test_transitivity_signed(): x = load_signed_sample(thres=.85) t = bct.transitivity_wu(x) assert np.imag(t)==0
# characteristic path length
cp = bct.charpath(thresh)
cp_s.append(cp[0])
# modularity
md = bct.modularity_louvain_und(thresh)
md_s.append(md[1])
# network measures of interest here
# global efficiency
at = bct.assortativity_wei(thresh)
at_s.append(at)
# modularity
tr = bct.transitivity_wu(thresh)
tr_s.append(tr)
df.at[(subject, session, task, conds[i], mask),
"assortativity"] = np.trapz(ge_s, dx=0.01)
df.at[(subject, session, task, conds[i], mask),
"transitivity"] = np.trapz(md_s, dx=0.01)
df.at[(subject, session, task, conds[i], mask),
"efficiency"] = np.trapz(ge_s, dx=0.01)
df.at[(subject, session, task, conds[i], mask),
"charpath"] = np.trapz(cp_s, dx=0.01)
df.at[(subject, session, task, conds[i], mask),
"modularity"] = np.trapz(md_s, dx=0.01)
# df.to_csv(join(sink_dir, 'resting-state_graphtheory_shen+craddock.csv'), sep=',')
lab_notebook.at[(subject, session, task, conds[i],
corr_pln += [CorrelationAnalyzer(nits)]
corr_cls_coef = [d.corrcoef for d in corr_cls]
pr_cls_tmp = np.asarray([
bct.centrality.pagerank_centrality(g, d=0.85)
for g in corr_cls_coef
]).mean(axis=0)
corr_pln_coef = [d.corrcoef for d in corr_pln]
pr_pln_tmp = np.asarray([
bct.centrality.pagerank_centrality(g, d=0.85)
for g in corr_pln_coef
]).mean(axis=0)
trans_cls_tmp = np.asarray(
[bct.transitivity_wu(g) for g in corr_cls_coef]).mean(axis=0)
trans_pln_tmp = np.asarray(
[bct.transitivity_wu(g) for g in corr_pln_coef]).mean(axis=0)
ge_cls_tmp = np.asarray(
[bct.distance.charpath(g)[1] for g in corr_cls_coef]).mean(
axis=0)
ge_pln_tmp = np.asarray(
[bct.distance.charpath(g)[1] for g in corr_pln_coef]).mean(
axis=0)
cp_cls_tmp = np.asarray(
[bct.distance.charpath(g)[0] for g in corr_cls_coef]).mean(
axis=0)
cp_pln_tmp = np.asarray(
[bct.distance.charpath(g)[0] for g in corr_pln_coef]).mean(
corr_pln += [CorrelationAnalyzer(nits)]
corr_cls_coef = [d.corrcoef for d in corr_cls]
pr_cls_tmp = np.asarray([
bct.centrality.pagerank_centrality(g, d=0.85)
for g in corr_cls_coef
]).mean(axis=0)
corr_pln_coef = [d.corrcoef for d in corr_pln]
pr_pln_tmp = np.asarray([
bct.centrality.pagerank_centrality(g, d=0.85)
for g in corr_pln_coef
]).mean(axis=0)
trans_cls_tmp = np.asarray(
[bct.transitivity_wu(g) for g in corr_cls_coef]).mean(axis=0)
trans_pln_tmp = np.asarray(
[bct.transitivity_wu(g) for g in corr_pln_coef]).mean(axis=0)
ge_cls_tmp = np.asarray([
bct.distance.charpath(g)[1] for g in corr_cls_coef
]).mean(axis=0)
ge_pln_tmp = np.asarray([
bct.distance.charpath(g)[1] for g in corr_pln_coef
]).mean(axis=0)
cp_cls_tmp = np.asarray([
bct.distance.charpath(g)[0] for g in corr_cls_coef
]).mean(axis=0)
cp_pln_tmp = np.asarray([
bct.distance.charpath(g)[0] for g in corr_pln_coef
def test_transitivity_wu():
x = load_sample(thres=.23)
t = bct.transitivity_wu(x)
assert np.allclose(t, 1.32927829)
def test_transitivity_signed():
x = load_signed_sample(thres=.85)
t = bct.transitivity_wu(x)
assert np.imag(t) == 0
def graph_estimates(cm, th):
#dictionary for storing our results
d = OrderedDict()
#thresholding moved here for other matrices than MatLab matrices
#removes negative weights
cm = bct.threshold_absolute(cm, 0.0)
cm = threshold_connected(cm, th)
#for binarizing the connectivity matrices,
#we work with weighted so this is turned off
#bin_cm = bct.binarize(cm)
#invert the connectivity for computing shortest paths
cm_inv = bct.invert(cm)
#modularity_und is found in modularity.py
modularity_und = bct.modularity_und(cm)
#the community_affiliation vector that gets input to some of the functions
community_affiliation = modularity_und[0]
#distance_wei and charpath is found in distance.py
distance_wei = bct.distance_wei(cm_inv)
charpath = bct.charpath(distance_wei[0], False, False)
#clustering_coef_wu is found in clustering.py
clustering_coef_wu = bct.clustering_coef_wu(cm)
avg_clustering_coef_wu = np.mean(clustering_coef_wu)
#assortativity_wei is found in core.py
d['assortativity_wei-r'] = bct.assortativity_wei(cm, flag=0)
#just taking the average of clustering_coef_wu
d['avg_clustering_coef_wu:C'] = avg_clustering_coef_wu
d['charpath-lambda'] = charpath[0]
#d['charpath-efficiency'] = charpath[1]
#d['charpath-ecc'] = charpath[2]
#d['charpath-radius'] = charpath[3]
#d['charpath-diameter'] = charpath[4]
d['clustering_coef_wu-C'] = clustering_coef_wu
d['efficiency_wei-Eglob'] = bct.efficiency_wei(cm)
#d['efficiency_wei-Eloc'] = bct.efficiency_wei(cm, True)
#d['modularity_und-ci'] = modularity_und[0]
d['modularity_und-Q'] = modularity_und[1]
d['small_worldness:S'] = compute_small_worldness(cm,
avg_clustering_coef_wu,
charpath[0])
#transitivity_wu can be found in clustering.py
d['transitivity_wu-T'] = bct.transitivity_wu(cm)
#EXAMPLES for local measures and binary measures. Comment in to use.
#VECTOR MEASURES
#d['betweenness_wei-BC'] = bct.betweenness_wei(cm_inv)
# d['module_degree_zscore-Z'] = bct.module_degree_zscore(cm, community_affiliation)
#d['degrees_und-deg'] = bct.degrees_und(cm)
#d['charpath-ecc'] = charpath[2]
#BINARIES
# d['clustering_coef_bu-C'] = bct.clustering_coef_bu(bin_cm)
# d['efficiency_bin-Eglob'] = bct.efficiency_bin(bin_cm)
# d['efficiency_bin-Eloc'] = bct.efficiency_bin(bin_cm, True)
# d['modularity_und_bin-ci'] = modularity_und_bin[0]
# d['modularity_und_bin-Q'] = modularity_und_bin[1]
# d['transitivity_bu-T'] = bct.transitivity_bu(bin_cm)
# d['betweenness_bin-BC'] = bct.betweenness_bin(bin_cm)
# modularity_und_bin = bct.modularity_und(bin_cm)
#d['participation_coef'] = bct.participation_coef(cm, community_affiliation)
######## charpath giving problems with ecc, radius and diameter
# np.seterr(invalid='ignore')
return d
for subject in subjects:
cls = np.load(source_folder + "graph_data/%s_classic_pow_pln.npy" %
subject).item()
pln = np.load(source_folder + "graph_data/%s_plan_pow_pln.npy" %
subject).item()
cls_all.append(cls)
pln_all.append(pln)
for k, band in enumerate(bands.keys()):
data_cls = []
for j in range(len(cls_all)):
tmp = cls_all[j][band]
data_cls.append(np.asarray([bct.transitivity_wu(g)
for g in tmp]).mean(axis=0))
data_pln = []
for j in range(len(pln_all)):
tmp = pln_all[j][band]
data_pln.append(np.asarray([bct.transitivity_wu(g)
for g in tmp]).mean(axis=0))
data_cls = np.asarray(data_cls)
data_pln = np.asarray(data_pln)
X = np.vstack([data_cls, data_pln])
y = np.concatenate([np.zeros(len(data_cls)), np.ones(len(data_pln))])
cv = StratifiedShuffleSplit(y, test_size=0.1)
for subject in subjects:
cls = np.load(source_folder +
"graph_data/%s_classic_pow_pln.npy" % subject).item()
pln = np.load(source_folder +
"graph_data/%s_plan_pow_pln.npy" % subject).item()
cls_all.append(cls)
pln_all.append(pln)
for k, band in enumerate(bands.keys()):
data_cls = []
for j in range(len(cls_all)):
tmp = cls_all[j][band]
data_cls.append(
np.asarray([bct.transitivity_wu(g) for g in tmp]).mean(axis=0))
data_pln = []
for j in range(len(pln_all)):
tmp = pln_all[j][band]
data_pln.append(
np.asarray([bct.transitivity_wu(g) for g in tmp]).mean(axis=0))
data_cls = np.asarray(data_cls)
data_pln = np.asarray(data_pln)
X = np.vstack([data_cls, data_pln])
y = np.concatenate([np.zeros(len(data_cls)), np.ones(len(data_pln))])
cv = StratifiedShuffleSplit(y, test_size=0.1)
model = joblib.load(source_folder +
for subject in subjects:
cls = np.load(source_folder + "graph_data/%s_classic_pow_post.npy" %
subject).item()
pln = np.load(source_folder + "graph_data/%s_plan_pow_post.npy" %
subject).item()
cls_all.append(cls)
pln_all.append(pln)
for k, band in enumerate(bands.keys()):
data_cls = []
for j in range(len(cls_all)):
tmp = cls_all[j][band]
data_cls.append(np.asarray([bct.transitivity_wu(g)
for g in tmp]).mean(axis=0))
data_pln = []
for j in range(len(pln_all)):
tmp = pln_all[j][band]
data_pln.append(np.asarray([bct.transitivity_wu(g)
for g in tmp]).mean(axis=0))
data_cls = np.asarray(data_cls)
data_pln = np.asarray(data_pln)
X = np.vstack([data_cls, data_pln])
y = np.concatenate([np.zeros(len(data_cls)), np.ones(len(data_pln))])
cv = StratifiedShuffleSplit(y, test_size=0.1)
def calc_graph_vector(filename, thresholds) :
'''
This function calculates graph measures for connectivity matrix loaded from textfile
and save results under the same name with additional superscript +'_GV' (in same dir
filename is located)
Input arguments:
filename(str): name of file containing connectivity matrix (txt extension)
thresholds(list): list containing thresholds of interest #
Kamil Bonna, 14.08.2018
'''
#--- check inputs
import os
if not os.path.exists(filename):
raise Exception('{} does not exist'.format(filename))
if type(thresholds) != list:
raise Exception('thresholds should be a list!')
import numpy as np
import bct
#=== inner variables
N_rep_louvain = 10 # number of Louvain algorithm repetitions
N_measures = 10 # number of graph measures
gamma = 1 # Louvain resolution parameter
#--- load matrix
A_raw = np.loadtxt(filename)
N = A_raw.shape[0] # number of nodes
M_sat = N*(N-1)/2 # max number of connections
#=== calculate output
graph_measures = np.zeros([ len(thresholds), N_measures ]) # create empty output matrix
for thr in range(len(thresholds)) :
#--- thresholding
A = bct.threshold_proportional( A_raw, p=thresholds[thr], copy=True );
A[np.nonzero(A<0)] = 0 # ensure only positive weights
M_act = A[np.nonzero(A>0)].shape[0] / 2 # actual number of nonzero connections
#--- calculate measures
#-- mean connection strenght
S = np.sum(A)/M_act
#-- connection strenght std
Svar = np.std(A[np.nonzero(A)])
#-- modularity
[M,Q] = bct.modularity_louvain_und(A, gamma)
for i in range(N_rep_louvain) :
[Mt,Qt] = bct.modularity_louvain_und(A, gamma)
if Qt > Q :
Q = Qt
M = Mt
#-- participation coefficient
P = np.mean(bct.participation_coef_sign(A, M))
#-- clustering
C = np.mean(bct.clustering_coef_wu(A))
#-- transitivity
T = bct.transitivity_wu(A)
#-- assortativity
Asso = bct.assortativity_wei(A)
#-- global & local efficiency
Eglo = bct.efficiency_wei(A)
Eloc = np.mean(bct.efficiency_wei(A, local=True))
#-- mean eigenvector centralit
Eig = np.mean(bct.eigenvector_centrality_und(A))
#--- write vector to matrix
graph_measures[thr] = [ S, Svar, Q, P, C, T, Asso, Eglo, Eloc, Eig ]
#=== save results to file
np.savetxt( filename[:-4]+'_GV.txt', graph_measures )
