"""Routines and class for Bogoliubov MBPT diagrams."""
from future import standard_library
standard_library.install_aliases()
from builtins import str
from builtins import zip
from builtins import range
from adg.tools import reversed_enumerate
import itertools
import numpy as np
import networkx as nx
import adg.tsd
import adg.diag
[docs]def diagrams_generation(p_order, three_body_use, nbody_obs, canonical):
"""Generate diagrams for BMBPT from bottom up.
Args:
p_order (int): The BMBPT perturbative order of the studied diagrams.
three_body_use (bool): Flag for the use of three-body forces.
nbody_obs (int): N-body character of the obervable of interest.
canonical (bool): ``True`` if one draws only canonical diagrams.
Returns:
(list): NumPy arrays encoding the adjacency matrices of the graphs.
>>> diags = diagrams_generation(1, False, 2, False)
>>> len(diags)
2
>>> any(np.array_equal([[0, 4], [0, 0]], diag) for diag in diags)
True
>>> any(np.array_equal([[0, 2], [0, 0]], diag) for diag in diags)
True
>>> diags = diagrams_generation(1, True, 3, False)
>>> len(diags)
3
>>> any(np.array_equal([[0, 6], [0, 0]], diag) for diag in diags)
True
>>> any(np.array_equal([[0, 4], [0, 0]], diag) for diag in diags)
True
>>> any(np.array_equal([[0, 2], [0, 0]], diag) for diag in diags)
True
>>> diags = diagrams_generation(2, False, 2, True)
>>> len(diags)
2
>>> any(np.array_equal([[0, 2, 2], [0, 0, 2], [0, 0, 0]], d) for d in diags)
True
>>> any(np.array_equal([[0, 1, 1], [0, 0, 3], [0, 0, 0]], d) for d in diags)
True
"""
# Matrices contain operator vertex + p_order perturbative vertices
order = p_order + 1
# Create a null oriented adjacency matrix of dimension (p_order,p_order)
matrices = [np.zeros((order, order), dtype=int)]
# Generate oriented adjacency matrices going vertex-wise
vertices = list(range(order))
add = matrices.append
for vertex in vertices:
if vertex == 0:
deg_max = 2*nbody_obs
else:
deg_max = 6 if three_body_use else 4
for sum_index in range(vertex+1, order):
for mat in reversed(matrices):
elem_max = deg_max - sum(mat[k][vertex] + mat[vertex][k]
for k in vertices)
for elem in range(1, elem_max + 1):
temp_mat = mat.copy()
temp_mat[vertex, sum_index] = elem
add(temp_mat)
adg.diag.check_vertex_degree(
matrices, three_body_use, nbody_obs, canonical, vertex
)
if 0 < vertex < order-1:
check_unconnected_spawn(matrices, vertex)
remove_disconnected_matrices(matrices)
return order_and_remove_topologically_equiv(matrices, order - 1)
[docs]def remove_disconnected_matrices(matrices):
"""Remove matrices corresponding to disconnected diagrams.
Args:
matrices (list): List of adjacency matrices.
"""
vertices = list(range(matrices[0].shape[0]))
permutations = [[0] + list(k)
for k in itertools.permutations(vertices[1:])]
for idx, matrix in reversed_enumerate(matrices):
for reordering in permutations:
mat = matrix[:, reordering][reordering, :]
for vert in vertices[1:]:
# Check if the permuted matrix is block-diagonal
if not mat[vertices[:vert], :][:, vertices[vert:]].any() \
and not mat[:, vertices[:vert]
][vertices[vert:], :].any():
del matrices[idx]
break
else:
continue
break
[docs]def order_and_remove_topologically_equiv(matrices, max_vertex):
"""Order the matrices in sub-list and remove topologically equivalent ones.
Args:
matrices (list): The adjacency matrices to be checked.
max_vertex (int): The maximum vertex which has been filled.
Returns:
(list): The ordered topologically unique matrices.
"""
matrices_dict = {}
for idx, matrix in reversed_enumerate(matrices):
row0 = "".join("%i" % elem for elem
in np.sort(matrix[0, :]).tolist())
if row0 in matrices_dict:
matrices_dict[row0].append(matrix)
else:
matrices_dict[row0] = [matrix]
del matrices[idx]
for row_key in sorted(matrices_dict.keys()):
matrices_dict[row_key] = \
check_topologically_equivalent(matrices_dict[row_key], max_vertex)
matrices = []
for matrices_list in list(matrices_dict.values()):
matrices += matrices_list
return matrices
[docs]def check_topologically_equivalent(matrices, max_vertex):
"""Exclude matrices that would spawn topologically equivalent graphs.
Args:
matrices (list): Adjacency matrices to be checked.
max_vertex (int): The maximum vertex which have been filled.
Returns:
(list): The topologically unique matrices.
>>> import numpy
>>> mats = [numpy.array([[0, 2, 0, 0], [2, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]]), \
numpy.array([[0, 0, 2, 0], [0, 0, 0, 0], [2, 0, 0, 1], [0, 0, 0, 0]])]
>>>
>>> mats = check_topologically_equivalent(mats, 2)
>>> mats # doctest: +NORMALIZE_WHITESPACE
[array([[0, 2, 0, 0], [2, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]])]
>>>
>>> mats = check_topologically_equivalent([], 2)
>>> mats # doctest: +NORMALIZE_WHITESPACE
[]
"""
if not matrices:
return []
vertices = list(range(matrices[0].shape[0]))
permutations = [[0] + list(k) + vertices[max_vertex+1:]
for k in itertools.permutations(vertices[1:max_vertex+1])]
for ind_mat1, mat1 in reversed_enumerate(matrices[:-1]):
mat1_1plus_sorted = np.sort(mat1[1:max_vertex+1, :].flat)
for ind_mat2 in range(len(matrices)-1, ind_mat1, -1):
mat2 = matrices[ind_mat2]
# Basic check to avoid needless permutations
if not (mat1_1plus_sorted
- np.sort(mat2[1:max_vertex+1, :].flat)).any():
# Test for all possible permutations
for reordering in permutations:
if not (mat1 - mat2[:, reordering][reordering, :]).any():
del matrices[ind_mat2]
break
else:
continue
break
return matrices
[docs]def check_unconnected_spawn(matrices, max_filled_vertex):
"""Exclude some matrices that would spawn unconnected diagrams.
Do several permutations among the rows and columns corresponding to already
filled vertices, and check if one obtains a block-diagonal organisation,
where the off-diagonals blocks connecting the already-filled and
yet-unfilled parts of the matrix would be empty. In that case, remove the
matrix.
Args:
matrices (list): The adjacency matrices to be checked.
max_filled_vertex (int): The furthest vertex until which the matrices
have been filled.
>>> import numpy
>>> mats = [numpy.array([[0, 2, 0], [2, 0, 0], [0, 0, 0]]), \
numpy.array([[0, 2, 1], [2, 0, 1], [0, 0, 0]])]
>>>
>>> check_unconnected_spawn(mats, 1)
>>> mats # doctest: +NORMALIZE_WHITESPACE
[array([[0, 2, 1], [2, 0, 1], [0, 0, 0]])]
"""
vertices = list(range(matrices[0].shape[0]))
permutations = [[0] + list(k) + vertices[max_filled_vertex+1:]
for k
in itertools.permutations(vertices[1:max_filled_vertex+1])]
for ind_mat, matrix in reversed_enumerate(matrices):
# Test for all possible permutations
for reordering in permutations:
mat = matrix[:, reordering][reordering, :]
# Check for non-zero elements in off-diagonal blocks
if not mat[:, vertices[:max_filled_vertex+1]
][vertices[max_filled_vertex+1:], :].any() \
and not mat[vertices[:max_filled_vertex+1],
:][:, vertices[max_filled_vertex+1:]].any():
del matrices[ind_mat]
break
[docs]def produce_expressions(diagrams, diagrams_time):
"""Produce and store the expressions associated to the BMBPT diagrams.
Args:
diagrams (list): The list of all BmbptFeynmanDiagrams.
diagrams_time (list): Their associates TSDs.
"""
for diag in diagrams:
diag.attribute_qp_labels()
for t_diag in diagrams_time:
if diag.unique_id in t_diag.tags[1:]:
diag.time_tag = t_diag.tags[0]
diag.tsd_is_tree = t_diag.is_tree
break
diag.attribute_expressions(diagrams_time[diag.time_tag])
[docs]def order_diagrams(diagrams):
"""Order the BMBPT diagrams and return number of diags for each type.
Args:
diagrams (list): Possibly redundant BmbptFeynmanDiagrams.
Returns:
(tuple): First element is the list of topologically unique, ordered
diagrams. Second element is a dict with the number of diagrams
for each major type. Third element is a dict with the identifiers
of diagrams starting each output file section.
"""
diagrams_2_hf = []
diagrams_2_ehf = []
diagrams_2_not_hf = []
diagrams_3_hf = []
diagrams_3_ehf = []
diagrams_3_not_hf = []
for i_diag, diag in reversed_enumerate(diagrams):
if diag.two_or_three_body == 2:
if diag.hf_type == "HF":
diagrams_2_hf.append(diag)
elif diag.hf_type == "EHF":
diagrams_2_ehf.append(diag)
elif diag.hf_type == "noHF":
diagrams_2_not_hf.append(diag)
elif diag.two_or_three_body == 3:
if diag.hf_type == "HF":
diagrams_3_hf.append(diag)
elif diag.hf_type == "EHF":
diagrams_3_ehf.append(diag)
elif diag.hf_type == "noHF":
diagrams_3_not_hf.append(diag)
del diagrams[i_diag]
diagrams = diagrams_2_hf + diagrams_2_ehf + diagrams_2_not_hf \
+ diagrams_3_hf + diagrams_3_ehf + diagrams_3_not_hf
diags_nb_per_type = {
'nb_2_hf': len(diagrams_2_hf),
'nb_2_ehf': len(diagrams_2_ehf),
'nb_2_not_hf': len(diagrams_2_not_hf),
'nb_3_hf': len(diagrams_3_hf),
'nb_3_ehf': len(diagrams_3_ehf),
'nb_3_not_hf': len(diagrams_3_not_hf),
'nb_diags': len(diagrams),
'nb_2': (len(diagrams_2_hf) + len(diagrams_2_ehf)
+ len(diagrams_2_not_hf)),
'nb_3': (len(diagrams_3_hf) + len(diagrams_3_ehf)
+ len(diagrams_3_not_hf))
}
section_flags = {
'two_body_hf': diagrams_2_hf[0].unique_id if diagrams_2_hf else -1,
'two_body_ehf': diagrams_2_ehf[0].unique_id if diagrams_2_ehf else -1,
'two_body_not_hf': diagrams_2_not_hf[0].unique_id
if diagrams_2_not_hf else -1,
'three_body_hf': diagrams_3_hf[0].unique_id if diagrams_3_hf else -1,
'three_body_ehf': diagrams_3_ehf[0].unique_id
if diagrams_3_ehf else -1,
'three_body_not_hf': diagrams_3_not_hf[0].unique_id
if diagrams_3_not_hf else -1
}
return diagrams, diags_nb_per_type, section_flags
[docs]class BmbptFeynmanDiagram(adg.diag.Diagram):
"""Describes a BMBPT Feynman diagram with its related properties.
Attributes:
two_or_three_body (int): The 2 or 3-body characted of the vertices.
time_tag (int): The tag number associated to the diagram's
associated TSD.
tsd_is_tree (bool): The tree or non-tree character of the
associated TSD.
feynman_exp (str): The Feynman expression associated to the diagram.
diag_exp (str): The Goldstone expression associated to the diagram.
vert_exp (list): The expression associated to the vertices.
hf_type (str): The Hartree-Fock, non-Hartree-Fock or Hartree-Fock for
the energy operator only character of the graph.
unique_id (int): A unique number associated to the diagram.
vertex_exchange_sym_factor (int): Lazy-initialized symmetry factor
associated to the vertex exchange, stored to avoid being computed
several times.
"""
__slots__ = ('two_or_three_body', 'time_tag', 'tsd_is_tree', 'feynman_exp',
'diag_exp', 'vert_exp', 'hf_type', 'unique_id',
'_vert_exchange_sym_fact')
def __init__(self, nx_graph, tag_num):
"""Generate a BMBPT diagrams using a NetworkX graph.
Args:
nx_graph (NetworkX MultiDiGraph): The graph of interest.
tag_num (int): The tag number associated to the diagram.
"""
adg.diag.Diagram.__init__(self, nx_graph)
self.two_or_three_body = 3 if self.max_degree == 6 else 2
self.tags = [tag_num]
self.time_tag = -1
self.tsd_is_tree = False
self.feynman_exp = ""
self.diag_exp = ""
self.vert_exp = []
if 2 not in self.degrees:
self.hf_type = "HF"
elif 2 not in self.unsort_degrees[1:]:
self.hf_type = "EHF"
else:
self.hf_type = "noHF"
self.unique_id = tag_num
self._vert_exchange_sym_fact = None
[docs] def attribute_expressions(self, time_diag):
"""Attribute the correct Feynman and Goldstone expressions.
Args:
time_diag (TimeStructureDiagram): The associated TSD.
"""
self.vert_exp = [self.vertex_expression(vertex)
for vertex in self.graph]
numerator = self.extract_numerator()
denominator = self.time_tree_denominator(
nx.relabel_nodes(time_diag.graph, time_diag.perms[self.unique_id])
) if self.tsd_is_tree else ""
extra_factor = "" if self.tsd_is_tree \
else "\\left[" \
+ " + ".join("\\frac{1}{%s}"
% self.time_tree_denominator(
nx.relabel_nodes(equi_t_graph,
time_diag.perms[self.unique_id]))
for equi_t_graph in time_diag.equivalent_trees) \
+ " \\right]"
# Determine the pre-factor
prefactor = "(-1)^%i " % (len(self.graph) - 1)
if self.has_sign_factor():
prefactor = "-%s" % prefactor
sym_fact = self.symmetry_factor()
prefactor = "\\frac{%s}{%s}\\sum_{k_i}" % (prefactor, sym_fact) \
if sym_fact != "" else "%s\\sum_{k_i}" % prefactor
# Set the Feynman and Goldstone expressions
self.feynman_exp = \
"\\lim\\limits_{\\tau \\to \\infty}%s%s\\int_{0}^{\\tau}%s\n" \
% (prefactor, numerator, self.extract_integral())
self.diag_exp = \
"%s\\frac{%s}{%s} %s\n" % (prefactor, numerator,
denominator, extra_factor) \
if denominator != "" \
else "%s%s%s\n" % (prefactor, numerator, extra_factor)
[docs] def has_sign_factor(self):
"""Return True if a sign factor is associated to the diagram.
Wrapper allowing for easy refactoring of expression code.
Returns:
(boolean): The presence of a sign factor.
"""
# Use exclusive or for the sign factor
return self.has_crossing_sign()
[docs] def symmetry_factor(self):
"""Return the overall symmetry factor of the diagram.
Returns:
(str): The combination of all symmetry factors.
"""
sym_factor = ""
for vertex_degrees in self.unsort_io_degrees:
if self.unsort_io_degrees.count(vertex_degrees) >= 2:
vertex_sym = self.vertex_exchange_sym_factor
sym_factor += "%i" % vertex_sym if vertex_sym > 1 else ""
break
sym_factor += self.multiplicity_symmetry_factor()
return sym_factor
[docs] def vertex_expression(self, vertex):
"""Return the expression associated to a given vertex.
Args:
vertex (int): The vertex of interest in the graph.
Returns:
(str): The LaTeX expression associated to the vertex.
"""
expression = r"\epsilon^{" \
+ "".join("%s"
% prop[3]['qp_state']
for prop
in self.graph.out_edges(vertex, keys=True, data=True)) \
+ "}_{" \
+ "".join("%s"
% prop[3]['qp_state']
for prop
in self.graph.in_edges(vertex, keys=True, data=True)) \
+ "}"
return expression
[docs] def write_graph(self, latex_file, directory, write_time):
"""Write the BMBPT graph and its associated TSD to the LaTeX file.
Args:
latex_file (file): The LaTeX output file of the program.
directory (str): The path to the result folder.
write_time (bool): ``True`` if we want informations on the
associated TSDs.
"""
latex_file.write('\n\\begin{center}\n')
adg.diag.draw_diagram(directory, latex_file, str(self.tags[0]), 'diag')
if write_time:
latex_file.write(
'\\hspace{10pt} $\\rightarrow$ \\hspace{10pt} T%i:'
% (self.time_tag + 1))
adg.diag.draw_diagram(directory, latex_file,
str(self.time_tag), 'time')
latex_file.write('\n\\end{center}\n\n')
[docs] def write_tsd_info(self, diagrams_time, latex_file):
"""Write info related to the BMBPT associated TSD to the LaTeX file.
Args:
diagrams_time (list): The associated TSDs.
latex_file (file): The LaTeX output file of the program.
"""
for tdiag in diagrams_time:
if self.time_tag == tdiag.tags[0]:
time_diag = tdiag
break
latex_file.write(
"\\begin{equation}\n\\text{T}%i = " % (self.time_tag + 1)
+ "%s\\end{equation}\n" % time_diag.expr)
self.write_vertices_values(latex_file, time_diag.perms[self.unique_id])
[docs] def write_section(self, result, commands, section_flags):
"""Write section and subsections for BMBPT result file.
Args:
result (file): The LaTeX output file of the program.
commands (dict): The flags associated with run management.
section_flags (dict): UniqueIDs of diags starting each section.
"""
if self.unique_id == section_flags['two_body_hf']:
result.write(
"\\section{Two-body diagrams}\n\n"
+ "\\subsection{Two-body energy canonical diagrams}\n\n")
elif self.unique_id == section_flags['two_body_ehf']:
result.write("\\subsection{Two-body canonical diagrams " +
"for a generic operator only}\n\n")
elif self.unique_id == section_flags['two_body_not_hf']:
result.write("\\subsection{Two-body non-canonical diagrams}\n\n")
if commands.with_3NF:
if self.unique_id == section_flags['three_body_hf']:
result.write(
"\\section{Three-body diagrams}\n\n"
+ "\\subsection{Three-body energy canonical diagrams}\n\n")
elif self.unique_id == section_flags['three_body_ehf']:
result.write("\\subsection{Three-body canonical diagrams " +
"for a generic operator only}\n\n")
elif self.unique_id == section_flags['three_body_not_hf']:
result.write(
"\\subsection{Three-body non-canonical diagrams}\n\n")
if commands.theory == "PBMBPT":
result.write("\\paragraph{Diagram %i.%i:}\n" % (self.tags[0] + 1,
self.tags[1] + 1))
else:
result.write("\\paragraph{Diagram %i:}\n" % (self.tags[0] + 1))
self.write_diag_exps(result, commands.order)
[docs] def write_vertices_values(self, latex_file, mapping):
"""Write the qp energies associated to each vertex of the diag.
Args:
latex_file (file): The LaTeX output file of the program.
mapping (dict): A mapping between the vertices in the diagram and
the vertices in its euivalent TSD, since permutations between
vertices are possible.
"""
latex_file.write("\\begin{align*}\n")
for ind in range(1, len(self.vert_exp)):
latex_file.write("a_%i &= %s" % (ind, self.vert_exp[mapping[ind]]))
if ind != len(self.vert_exp)-1:
latex_file.write(r"\\")
latex_file.write('\n')
latex_file.write("\\end{align*}\n")
[docs] def write_diag_exps(self, latex_file, norder):
"""Write the expressions associated to a diagram in the LaTeX file.
Args:
latex_file (file): The LaTeX outputfile of the program.
norder (int): The order in BMBPT formalism.
"""
latex_file.write(
"\\begin{align}\n\\text{PO}%i.%i\n" % (norder, (self.tags[0] + 1))
+ "&= %s" % self.feynman_exp
+ r" \nonumber \\" + "\n"
+ "&= %s\\end{align}\n" % self.diag_exp)
@property
def vertex_exchange_sym_factor(self):
"""Return the symmetry factor associated with vertex exchange.
Returns:
(int): The symmetry factor for vertex exchange.
"""
if self._vert_exchange_sym_fact is None:
factor = len(self.equivalent_permutations())
self._vert_exchange_sym_fact = max(factor, 1)
return self._vert_exchange_sym_fact
[docs] def equivalent_permutations(self):
"""Return the permutations generating equivalent diagrams.
Returns:
(list): Vertices permutations as dictionnaries.
"""
perm_vertices = [vertex for vertex, degrees
in enumerate(self.unsort_io_degrees)
if not self.graph.nodes[vertex]['operator']
and self.unsort_io_degrees.count(degrees) >= 2]
permutations = []
op_nm = nx.algorithms.isomorphism.categorical_node_match('operator',
False)
for permutation in itertools.permutations(perm_vertices):
permuted_graph = nx.relabel_nodes(self.graph,
dict(list(zip(perm_vertices,
permutation))),
copy=True)
# Check for a permutation that leaves the graph unchanged
# (only way to keep the edge list of the same length)
# (is_isomorphic could maybe be replaced by something faster here)
if nx.is_isomorphic(self.graph,
nx.intersection(self.graph, permuted_graph),
node_match=op_nm):
permutations.append(dict(list(zip(perm_vertices,
permutation))))
return permutations
[docs] def extract_integral(self):
"""Return the integral part of the Feynman expression of the diag.
Returns:
(str): The integral part of its Feynman expression.
"""
pert_vertex_indices = list(range(1, len(self.graph)))
integral = "".join("\\mathrm{d}\\tau_%i" % vertex
for vertex in pert_vertex_indices)
if len(pert_vertex_indices) > 1:
for vertex_i in pert_vertex_indices:
integral += "".join("\\theta(\\tau_%i-\\tau_%i) " % (vertex_j,
vertex_i)
for vertex_j in pert_vertex_indices
if self.graph.has_edge(vertex_i, vertex_j))
integral += "".join("e^{-\\tau_%i %s}"
% (vertex, self.vert_exp[vertex])
for vertex in pert_vertex_indices)
return integral
[docs] def attribute_qp_labels(self):
"""Attribute the appropriate qp labels to the graph's propagators."""
for idx, prop in enumerate(self.graph.edges(keys=True, data=True)):
prop[3]['qp_state'] = "k_{%i}" % (idx+1)
[docs] def extract_numerator(self):
"""Return the numerator associated to a BMBPT graph.
Returns:
(str): The numerator of the graph.
"""
graph = self.graph
numerator = ""
for vertex in graph:
# Attribute the correct operator to each vertex
numerator += "O" if graph.nodes[vertex]['operator'] else "\\Omega"
# Attribute the good "type number" to each vertex
numerator += "^{%i%i}_{" % (self.unsort_io_degrees[vertex][1],
self.unsort_io_degrees[vertex][0])
# First add the qp states corresponding to propagators going out
numerator += "".join(prop[3]['qp_state']
for prop
in graph.out_edges(vertex,
keys=True, data=True))
# Add the qp states corresponding to propagators coming in
for previous_vertex in reversed(range(vertex)):
numerator += "".join(
prop[3]['qp_state']
for prop in graph.in_edges(vertex, keys=True, data=True)
if prop[0] == previous_vertex)
numerator += "} "
return numerator
[docs] def has_crossing_sign(self):
"""Return True for a minus sign associated with crossing propagators.
Use the fact that all lines propagate upwards and the
canonical representation of the diagrams and vertices.
Returns:
(bool): Encode for the sign factor associated with crossing
propagators.
"""
nb_crossings = 0
for vertex in self.graph:
for propagator in self.graph.out_edges(vertex, keys=True):
for vertex_ante in range(propagator[0]):
for vertex_post in range(propagator[0]+1, propagator[1]):
nb_crossings += self.graph.number_of_edges(vertex_ante,
vertex_post)
return nb_crossings % 2 == 1
[docs] def multiplicity_symmetry_factor(self):
"""Return the symmetry factor associated with propagators multiplicity.
Returns:
(str): The symmetry factor associated with equivalent lines.
"""
factor = ""
# Account for up to three-body operators
prop_multiplicity = [0 for _ in range(6)]
for vertex_i in self.graph:
for vertex_j in self.graph:
if self.graph.number_of_edges(vertex_i, vertex_j) >= 2:
prop_multiplicity[self.graph.number_of_edges(
vertex_i, vertex_j) - 1] += 1
for prop_id, multiplicity in enumerate(prop_multiplicity):
if multiplicity == 1:
factor += "(%i!)" % (prop_id+1)
elif multiplicity >= 2:
factor += "(%i!)" % (prop_id+1) + "^%i" % multiplicity
return factor
[docs] def time_tree_denominator(self, time_graph):
"""Return the denominator for a time-tree graph.
Args:
time_graph (NetworkX MultiDiGraph): Its associated time-structure
graph.
Returns:
(str): The denominator of the graph.
"""
denominator = ""
for vertex_i in range(1, len(time_graph)):
subgraph_stack = [vertex_j for vertex_j
in nx.descendants(time_graph, vertex_i)]
subgraph_stack.append(vertex_i)
subdiag = self.graph.subgraph(subgraph_stack)
denominator += "%s\\ " % adg.diag.extract_denom(self.graph,
subdiag)
return denominator