def graph_sparsify(M, epsilon, maxiter=10):
r"""
Sparsify a graph using Spielman-Srivastava algorithm.
Parameters
----------
M : Graph or sparse matrix
Graph structure or a Laplacian matrix
epsilon : int
Sparsification parameter
Returns
-------
Mnew : Graph or sparse matrix
New graph structure or sparse matrix
Note
----
Epsilon should be between 1/sqrt(N) and 1
Examples
--------
>>> from pygsp import graphs, operators
>>> G = graphs.Sensor(256, Nc=20, distribute=True)
>>> epsilon = 0.4
>>> G2 = operators.graph_sparsify(G, epsilon)
Reference
---------
See :cite: `spielman2011graph` `rudelson1999random` `rudelson2007sampling`
for more informations
"""
# Test the input parameters
if isinstance(M, Graph):
if not M.lap_type == 'combinatorial':
raise NotImplementedError
L = M.L
else:
L = M
N = np.shape(L)[0]
if not 1./np.sqrt(N) <= epsilon < 1:
raise ValueError('GRAPH_SPARSIFY: Epsilon out of required range')
# pas sparse
resistance_distances = resistance_distance(L).toarray()
# Get the Weight matrix
if isinstance(M, Graph):
W = M.W
else:
W = np.diag(L.diagonal()) - L.toarray()
W[W < 1e-10] = 0
W = sparse.csc_matrix(W)
start_nodes, end_nodes, weights = sparse.find(sparse.tril(W))
# Calculate the new weights.
weights = np.maximum(0, weights)
Re = np.maximum(0, resistance_distances[start_nodes, end_nodes])
Pe = weights * Re
Pe = Pe / np.sum(Pe)
for i in range(maxiter):
# Rudelson, 1996 Random Vectors in the Isotropic Position
# (too hard to figure out actual C0)
C0 = 1 / 30.
# Rudelson and Vershynin, 2007, Thm. 3.1
C = 4 * C0
q = round(N * np.log(N) * 9 * C**2 / (epsilon**2))
results = stats.rv_discrete(values=(np.arange(np.shape(Pe)[0]), Pe)).rvs(size=q)
spin_counts = stats.itemfreq(results).astype(int)
per_spin_weights = weights / (q * Pe)
counts = np.zeros(np.shape(weights)[0])
counts[spin_counts[:, 0]] = spin_counts[:, 1]
new_weights = counts * per_spin_weights
sparserW = sparse.csc_matrix((new_weights, (start_nodes, end_nodes)),
shape=(N, N))
sparserW = sparserW + sparserW.T
sparserL = sparse.diags(sparserW.diagonal(), 0) - sparserW
if Graph(W=sparserW).is_connected():
break
elif i == maxiter - 1:
logger.warning('Despite attempts to reduce epsilon, sparsified graph is disconnected')
else:
epsilon -= (epsilon - 1/np.sqrt(N)) / 2.
if isinstance(M, Graph):
sparserW = sparse.diags(sparserL.diagonal(), 0) - sparserL
if not M.directed:
sparserW = (sparserW + sparserW.T) / 2.
Mnew = Graph(W=sparserW)
M.copy_graph_attributes(Mnew)
else:
Mnew = sparse.lil_matrix(sparserL)
return Mnew
def graph_sparsify(M, epsilon, maxiter=10):
r"""Sparsify a graph (with Spielman-Srivastava).
Parameters
----------
M : Graph or sparse matrix
Graph structure or a Laplacian matrix
epsilon : int
Sparsification parameter
Returns
-------
Mnew : Graph or sparse matrix
New graph structure or sparse matrix
Notes
-----
Epsilon should be between 1/sqrt(N) and 1
Examples
--------
>>> from pygsp import reduction
>>> G = graphs.Sensor(256, Nc=20, distributed=True)
>>> epsilon = 0.4
>>> G2 = reduction.graph_sparsify(G, epsilon)
References
----------
See :cite:`spielman2011graph`, :cite:`rudelson1999random` and :cite:`rudelson2007sampling`.
for more informations
"""
# Test the input parameters
if isinstance(M, graphs.Graph):
if not M.lap_type == 'combinatorial':
raise NotImplementedError
L = M.L
else:
L = M
N = np.shape(L)[0]
if not 1./np.sqrt(N) <= epsilon < 1:
raise ValueError('GRAPH_SPARSIFY: Epsilon out of required range')
# Not sparse
resistance_distances = utils.resistance_distance(L).toarray()
# Get the Weight matrix
if isinstance(M, graphs.Graph):
W = M.W
else:
W = np.diag(L.diagonal()) - L.toarray()
W[W < 1e-10] = 0
W = sparse.coo_matrix(W)
W.data[W.data < 1e-10] = 0
W = W.tocsc()
W.eliminate_zeros()
start_nodes, end_nodes, weights = sparse.find(sparse.tril(W))
# Calculate the new weights.
weights = np.maximum(0, weights)
Re = np.maximum(0, resistance_distances[start_nodes, end_nodes])
Pe = weights * Re
Pe = Pe / np.sum(Pe)
for i in range(maxiter):
# Rudelson, 1996 Random Vectors in the Isotropic Position
# (too hard to figure out actual C0)
C0 = 1 / 30.
# Rudelson and Vershynin, 2007, Thm. 3.1
C = 4 * C0
q = round(N * np.log(N) * 9 * C**2 / (epsilon**2))
results = stats.rv_discrete(values=(np.arange(np.shape(Pe)[0]), Pe)).rvs(size=int(q))
spin_counts = stats.itemfreq(results).astype(int)
per_spin_weights = weights / (q * Pe)
counts = np.zeros(np.shape(weights)[0])
counts[spin_counts[:, 0]] = spin_counts[:, 1]
new_weights = counts * per_spin_weights
sparserW = sparse.csc_matrix((new_weights, (start_nodes, end_nodes)),
shape=(N, N))
sparserW = sparserW + sparserW.T
sparserL = sparse.diags(sparserW.diagonal(), 0) - sparserW
if graphs.Graph(W=sparserW).is_connected():
break
elif i == maxiter - 1:
logger.warning('Despite attempts to reduce epsilon, sparsified graph is disconnected')
else:
epsilon -= (epsilon - 1/np.sqrt(N)) / 2.
if isinstance(M, graphs.Graph):
sparserW = sparse.diags(sparserL.diagonal(), 0) - sparserL
if not M.is_directed():
sparserW = (sparserW + sparserW.T) / 2.
Mnew = graphs.Graph(W=sparserW)
#M.copy_graph_attributes(Mnew)
else:
Mnew = sparse.lil_matrix(sparserL)
return Mnew
def sparsifyGraph(M, epsilon, maxiter=20):
# Test the input parameters
if isinstance(M, graphs.Graph):
L = M.L
else:
L = M
N = np.shape(L)[0]
if not 1. / np.sqrt(N) <= epsilon < 1:
raise ValueError('sparsifyGraph: Epsilon out of required range')
# Not sparse
resistance_distances = utils.resistance_distance(L).toarray()
# Get the Weight matrix
if isinstance(M, graphs.Graph):
W = M.W
else:
W = np.diag(L.diagonal()) - L.toarray()
W[W < 1e-10] = 0
W = sparse.coo_matrix(W)
W.data[W.data < 1e-10] = 0
W = W.tocsc()
W.eliminate_zeros()
start_nodes, end_nodes, weights = sparse.find(sparse.tril(W))
# Calculate the new weights
weights = np.maximum(0, weights)
Re = np.maximum(0, resistance_distances[start_nodes, end_nodes])
Pe = weights * Re
Pe = Pe / np.sum(Pe)
for i in range(maxiter):
C0 = 1 / 30.
C = 4 * C0
q = round(N * np.log(N) * 9 * C**2 / (epsilon**2))
results = stats.rv_discrete(values=(np.arange(np.shape(Pe)[0]),
Pe)).rvs(size=int(q))
spin_counts = stats.itemfreq(results).astype(int)
per_spin_weights = weights / (q * Pe)
counts = np.zeros(np.shape(weights)[0])
counts[spin_counts[:, 0]] = spin_counts[:, 1]
new_weights = counts * per_spin_weights
sparserW = sparse.csc_matrix((new_weights, (start_nodes, end_nodes)),
shape=(N, N))
sparserW = sparserW + sparserW.T
sparserL = sparse.diags(sparserW.diagonal(), 0) - sparserW
if graphs.Graph(sparserW).is_connected():
break
elif i == maxiter - 1:
logger.warning(
'Despite attempts to reduce epsilon, sparsified graph is disconnected'
)
else:
epsilon -= (epsilon - 1 / np.sqrt(N)) / 2.
if isinstance(M, graphs.Graph):
sparserW = sparse.diags(sparserL.diagonal(), 0) - sparserL
sparserW = (sparserW + sparserW.T) / 2.
Mnew = graphs.Graph(sparserW, coords=M.coords)
else:
Mnew = sparse.lil_matrix(sparserL)
return Mnew
def graph_sparsify(M, epsilon, maxiter=10):
from pygsp import utils
from scipy import sparse, stats
# Test the input parameters
if isinstance(M, graphs.Graph):
if not M.lap_type == 'combinatorial':
raise NotImplementedError
L = M.L
else:
L = M
N = np.shape(L)[0]
if not 1. / np.sqrt(N) <= epsilon < 1:
raise ValueError('GRAPH_SPARSIFY: Epsilon out of required range')
# Not sparse
resistance_distances = utils.resistance_distance(L).toarray()
# Get the Weight matrix
if isinstance(M, graphs.Graph):
W = M.W
else:
W = np.diag(L.diagonal()) - L.toarray()
W[W < 1e-10] = 0
W = sparse.coo_matrix(W)
W.data[W.data < 1e-10] = 0
W = W.tocsc()
W.eliminate_zeros()
start_nodes, end_nodes, weights = sparse.find(sparse.tril(W))
# Calculate the new weights.
weights = np.maximum(0, weights)
Re = np.maximum(0, resistance_distances[start_nodes, end_nodes])
Pe = weights * Re + 1e-4
Pe = Pe / np.sum(Pe)
for i in range(maxiter):
# Rudelson, 1996 Random Vectors in the Isotropic Position
# (too hard to figure out actual C0)
C0 = 1 / 30.
# Rudelson and Vershynin, 2007, Thm. 3.1
C = 4 * C0
q = round(N * np.log(N) * 9 * C**2 / (epsilon**2))
results = stats.rv_discrete(values=(np.arange(np.shape(Pe)[0]),
Pe)).rvs(size=int(q))
spin_counts = stats.itemfreq(results).astype(int)
per_spin_weights = weights / (q * Pe)
counts = np.zeros(np.shape(weights)[0])
counts[spin_counts[:, 0]] = spin_counts[:, 1]
new_weights = counts * per_spin_weights
sparserW = sparse.csc_matrix((new_weights, (start_nodes, end_nodes)),
shape=(N, N))
sparserW = sparserW + sparserW.T
sparserL = sparse.diags(sparserW.diagonal(), 0) - sparserW
# if graphs.Graph(W=sparserW).is_connected():
# break
# elif i == maxiter - 1:
# print('Despite attempts to reduce epsilon, sparsified graph is disconnected')
# else:
# epsilon -= (epsilon - 1/np.sqrt(N)) / 2.
if isinstance(M, graphs.Graph):
sparserW = sparse.diags(sparserL.diagonal(), 0) - sparserL
if not M.is_directed():
sparserW = (sparserW + sparserW.T) / 2.
Mnew = graphs.Graph(W=sparserW)
#M.copy_graph_attributes(Mnew)
else:
Mnew = sparse.lil_matrix(sparserL)
return Mnew
