def freq_single(QN, A,B,C,DJ,DJK,DK,subdj,subdk): Jupper = QN[0] ind_upper = Jupper - QN[2] + QN[1] Jlower = QN[3] ind_lower = Jlower - QN[5] + QN[4] H_upper = ham(Jupper, A,B,C,DJ,DJK,DK,subdj,subdk) E_upper = la.eigs(H_upper) E_upper = sort(E_upper) H_lower = ham(Jlower, A,B,C,DJ,DJK,DK,subdj,subdk) E_lower = la.eigs(H_lower) E_lower = sort(E_lower) return float(E_upper[ind_upper]-E_lower[ind_lower])
def sig_lmc(C, A):
'''
This a function that using Lumped Markov chain to calculate
the significance of clusters in a give communinity structure.
refer to "Piccardi 2011 in PloS one".
Here we normalize the original definition of persistence by
the size of the corresponding cluster to get a better
INPUT:
"A" is a N-by-N weighted adjacency matrix
"C" is a N-by-1 partition(cluster) vector
OUTPUT:
normalized persistence probability of all clusters
'''
'''
Transition Matrix
'''
C = np.asarray(C)
A = np.double(A)
P = np.linalg.solve(np.diag(np.sum(A,axis = 1)),A)
[eval, evec] = linalg.eigs(P.T, 1)
if min(evec)<0:
evec = -evec
pi = np.double(evec.T)
num_node = np.double(np.shape(A)[0])
cl_label = np.double(np.unique(C))
num_cl = len(cl_label)
H = np.zeros((num_node, num_cl),dtype = np.double)
for i in range(num_cl):
H[:, i] = np.double((C==cl_label[i]))
# Transition matrix of the lumped Markov chain
Q = np.dot(np.dot(np.dot(np.linalg.solve(np.diag(np.dot(pi,H).flatten()),H.T),np.diag(pi.flatten())),P),H)
persistence = np.multiply(np.divide(np.diag(Q), np.sum(H,axis = 0)),np.sum(H))
return persistence
def sig_lmc(C, A):
'''
This a function that using Lumped Markov chain to calculate
the significance of clusters in a give communinity structure.
refer to "Piccardi 2011 in PloS one".
Here we normalize the original definition of persistence by
the size of the corresponding cluster to get a better
INPUT:
"A" is a N-by-N weighted adjacency matrix
"C" is a N-by-1 partition(cluster) vector
OUTPUT:
normalized persistence probability of all clusters
'''
'''
Transition Matrix
'''
C = np.asarray(C)
A = np.double(A)
P = np.linalg.solve(np.diag(np.sum(A,axis = 1)),A)
[eval, evec] = linalg.eigs(P.T, 1)
if min(evec)<0:
evec = -evec
pi = np.double(evec.T)
num_node = np.double(np.shape(A)[0])
cl_label = np.double(np.unique(C))
num_cl = len(cl_label)
H = np.zeros((num_node, num_cl),dtype = np.double)
for i in range(num_cl):
H[:, i] = np.double((C==cl_label[i]))
# Transition matrix of the lumped Markov chain
Q = np.dot(np.dot(np.dot(np.linalg.solve(np.diag(np.dot(pi,H).flatten()),H.T),np.diag(pi.flatten())),P),H)
persistence = np.multiply(np.divide(np.diag(Q), np.sum(H,axis = 0)),np.sum(H))
return persistence
def sd_convergence(A, b, x0, x):
'''
Study convergence of steepest descent method based on x0 and condition
number of A.
'''
assert A.shape == (2, 2), 'A is not a 2x2 matrix'
assert b.shape == (2, ), 'b is not a vector of len 2'
assert x0.shape == (2, ), 'x0 is not a vector of len 2'
assert x.shape == (2, ), 'x0 is not a vector of len 2'
assert abs(A[0, 1] - A[1, 0]) < 1E-15, 'A is not positive definite'
lambdas, vecs = la.eigs(A)
assert np.all(lmabdas > 0), 'A is not positive definite'
lambda0, lambda1 = lambdas
u0, u1 = vecs[:, 0], vecs[:, 1]
# Swap if not sorted
if lambda0 > lambda1:
lambda1, lambd0 = lambda0, lambda1
u1, u0 = u0, u1
# error
e0 = x0 - x
k = lambda1/lambda0
mu = u0.dot(e0)/u1.dot(e0)
print 'Condition number is', k
print 'Initial guess yields mu', mu
X, n_iters = solve(A, b, x0)
print 'error in solution', la.norm(X - x)
print n_iters
