"""Base class for undirected simplicial complexes.
The SimplicialComplex class allows any hashable object as a node and can associate
key/value attribute pairs with each undirected simplex and node.
Multi-simplices are not allowed.
"""
from collections.abc import Hashable, Iterable
from copy import copy, deepcopy
from itertools import combinations, count
from warnings import warn
from ..exception import XGIError, frozen
from ..utils.utilities import powerset, update_uid_counter
from .hypergraph import Hypergraph
from .views import EdgeView, NodeView
__all__ = ["SimplicialComplex"]
[docs]class SimplicialComplex(Hypergraph):
r"""A class to represent undirected simplicial complexes.
A simplicial complex is a collection of subsets of a set of *nodes* or *vertices*.
It is a pair :math:`(V, E)`, where :math:`V` is a set of elements called *nodes* or
*vertices*, and :math:`E` is a set whose elements are subsets of :math:`V`, that is,
each :math:`e \in E` satisfies :math:`e \subset V`. The elements of :math:`E` are
called *simplices*. Additionally, if a simplex is part of a simplicial complex, all
its faces must be too. This makes simplicial complexes a special case of
hypergraphs.
The SimplicialComplex class allows any hashable object as a node and can associate
attributes to each node, simplex, or the simplicial complex itself, in the form of
key/value pairs.
Parameters
----------
incoming_data : input simplicial complex data, optional
Data to initialize the simplicial complex. If None (default), an empty
simplicial complex is created, i.e. one with no nodes or simplices. The data
can be in the following formats:
* simplex list
* simplex dictionary
* 2-column Pandas dataframe (bipartite edges)
* Incidence matrix: numpy ndarray or scipy.sparse array
* SimplicialComplex object
* Hypergraph object
**attr : dict, optional
Attributes to add to the simplicial complex as key, value pairs.
By default, None.
See Also
--------
~xgi.core.hypergraph.Hypergraph
Notes
-----
Unique IDs are assigned to each node and simplex internally and are used to refer to
them throughout.
The `attr` keyword arguments are added as simplicial complex attributes. To add node
or simplex attributes see :meth:`add_node` and :meth:`add_simplex`. Methods such as
:meth:`add_simplex` replace :class:`Hypergraph` methods such as :meth:`add_edge`
which here raise an error.
Examples
--------
>>> import xgi
>>> S = xgi.SimplicialComplex([[1, 2, 3], [4], [5, 6], [6, 7, 8]])
>>> S.nodes
NodeView((1, 2, 3, 4, 5, 6, 7, 8))
>>> S.edges
EdgeView((0, 1, 2, 3, 4, 5, 6, 7, 8, 9))
"""
def __init__(self, incoming_data=None, **attr):
self._edge_uid = count()
self._net_attr = self._net_attr_dict_factory()
self._node = self._node_dict_factory()
self._node_attr = self._node_attr_dict_factory()
self._edge = self._edge_dict_factory()
self._edge_attr = self._edge_attr_dict_factory()
self._nodeview = NodeView(self)
self._edgeview = EdgeView(self)
if incoming_data is not None:
from ..convert import to_simplicial_complex
to_simplicial_complex(incoming_data, create_using=self)
self._net_attr.update(attr) # must be after convert
def __str__(self):
"""Returns a short summary of the simplicial complex.
Returns
-------
string
SimplicialComplex information
Examples
--------
>>> import xgi
>>> S = xgi.SimplicialComplex(name="foo")
>>> str(S)
"SimplicialComplex named 'foo' with 0 nodes and 0 simplices"
"""
try:
return (
f"{type(self).__name__} named '{self['name']}' "
f"with {self.num_nodes} nodes and {self.num_edges} simplices"
)
except XGIError:
return (
f"Unnamed {type(self).__name__} with {self.num_nodes}"
f" nodes and {self.num_edges} simplices"
)
[docs] def add_edge(self, edge, idx=None, **attr):
"""Deprecated in SimplicialComplex. Use add_simplex instead"""
warn("add_edge is deprecated in SimplicialComplex. Use add_simplex instead")
return self.add_simplex(edge, idx=None, **attr)
[docs] def add_edges_from(self, ebunch_to_add, max_order=None, **attr):
"""Deprecated in SimplicialComplex. Use add_simplices_from instead"""
warn(
"add_edges_from is deprecated in SimplicialComplex. "
"Use add_simplices_from instead"
)
return self.add_simplices_from(ebunch_to_add, max_order=None, **attr)
[docs] def add_weighted_edges_from(
self, ebunch_to_add, max_order=None, weight="weight", **attr
):
"""Deprecated in SimplicialComplex. Use add_weighted_simplices_from instead"""
warn(
"add_weighted_edges_from is deprecated in SimplicialComplex."
" Use add_weighted_simplices_from instead"
)
return self.add_weighted_simplices_from(
ebunch_to_add, max_order=max_order, weight=weight, **attr
)
[docs] def remove_edge(self, idx):
"""Deprecated in SimplicialComplex. Use remove_simplex_id instead"""
warn(
"remove_edge is deprecated in SimplicialComplex. "
"Use remove_simplex_id instead"
)
return self.remove_simplex_id(idx)
[docs] def remove_edges_from(self, ebunch):
"""Deprecated in SimplicialComplex. Use remove_simplex_ids_from instead"""
warn(
"remove_edges_from is deprecated in SimplicialComplex. "
"Use remove_simplex_ids_from instead"
)
return self.remove_simplex_ids_from(ebunch)
[docs] def remove_node(self, n):
"""Remove a single node.
The removal is strong meaning that all edges containing the
node are removed.
Parameters
----------
n : node
A node in the simplicial complex
Raises
------
XGIError
If n is not in the simplicial complex.
See Also
--------
remove_nodes_from
"""
edge_neighbors = self._node[n]
del self._node[n]
del self._node_attr[n]
for e in edge_neighbors:
node_neighbors = self._edge[e]
del self._edge[e]
del self._edge_attr[e]
for node in node_neighbors.difference({n}):
self._node[node].remove(e)
[docs] def remove_nodes_from(self, nodes):
"""Remove multiple nodes.
Parameters
----------
nodes : iterable
An iterable of nodes.
See Also
--------
remove_node
"""
for n in nodes:
if n not in self:
warn(f"Node {n} not in simplicial complex")
continue
self.remove_node(n)
[docs] def add_node_to_edge(self, edge, node):
"""add_node_to_edge is not implemented in SimplicialComplex."""
raise XGIError("add_node_to_edge is not implemented in SimplicialComplex.")
def _add_simplex(self, members, idx=None, **attr):
"""Helper function to add a simplex to a simplicial complex, without any
check. Does not automatically update self._edge_uid"""
self._edge[idx] = set()
for node in members:
if node not in self._node:
if node is None:
raise ValueError("None cannot be a node")
self._node[node] = set()
self._node_attr[node] = self._node_attr_dict_factory()
self._node[node].add(idx)
self._edge[idx] = members
self._edge_attr[idx] = self._edge_attr_dict_factory()
self._edge_attr[idx].update(attr)
def _add_face(self, members):
"""Helper function to add a face to a simplicial complex, without any
check, and without attributes. Automatically updates self._edge_uid"""
idx = next(self._edge_uid)
self._edge[idx] = frozenset(members)
for n in members:
if n not in self._node:
self._node[n] = set()
self._node_attr[n] = self._node_attr_dict_factory()
self._node[n].add(idx)
self._edge_attr[idx] = self._edge_attr_dict_factory()
[docs] def add_simplex(self, members, idx=None, **attr):
"""Add a simplex to the simplicial complex, and all its subfaces that do
not exist yet.
Simplex attributes can be specified with keywords or by directly
accessing the simplex's attribute dictionary. The attributes do not propagate
to the subfaces.
Parameters
----------
members : Iterable
An iterable of the ids of the nodes contained in the new simplex.
idx : hashable, optional
Id of the new simplex. If None (default), a unique numeric ID will be
created.
**attr : dict, optional
Attributes of the new simplex.
See Also
--------
add_simplices_from : Add a collection of simplices.
Notes
-----
Currently cannot add empty simplices.
Examples
--------
Add simplices with or without specifying an simplex idx.
>>> import xgi
>>> S = xgi.SimplicialComplex()
>>> S.add_simplex([1, 2, 3])
>>> S.edges.members() # doctest: +NORMALIZE_WHITESPACE +SKIP
[frozenset({1, 2, 3}), frozenset({2, 3}),
frozenset({1, 2}), frozenset({1, 3})]
>>> S.add_simplex([3, 4], idx='myedge')
>>> S.edges
EdgeView((0, 1, 2, 3, 'myedge'))
Access attributes using square brackets. By default no attributes are created.
>>> S.edges[0]
{}
>>> S.add_simplex([1, 4], color='red', place='peru')
>>> S.edges
EdgeView((0, 1, 2, 3, 'myedge', 4))
>>> S.edges[4]
{'color': 'red', 'place': 'peru'}
"""
try:
members = frozenset(members)
except TypeError:
raise XGIError("The simplex cannot be cast to a frozenset.")
if self.has_simplex(members):
return
if idx in self._edge.keys(): # check that uid is not present yet
warn(f"uid {idx} already exists, cannot add simplex {members}")
return
idx = next(self._edge_uid) if not idx else idx
self._add_simplex(members, idx, **attr)
# set self._edge_uid correctly
update_uid_counter(self, idx)
# add all subfaces
faces = self._subfaces(members)
faces = set(faces) # get unique faces
for members_sub in faces:
# check that it does not exist yet (based on members, not ID)
if not members_sub or self.has_simplex(members_sub):
continue
self._add_face(members_sub)
def _subfaces(self, simplex, all=True):
"""Returns list of subfaces of simplex.
Parameters
----------
simplex: an iterable of hashables
A list of node ids
all: bool, optional
Whether to return all of the subfaces or just
those of the order below. By default, True.
Returns
-------
faces: list of iterables
The list containing the subfaces of the
given simplex
"""
size = len(simplex)
faces = []
if all:
for n in range(size, 2, -1):
for face in combinations(simplex, n - 1):
faces.append(face)
else:
for face in combinations(simplex, size - 1):
faces.append(face)
return faces
def _supfaces(self, simplex):
"""Returns list of simplices that contain simplex"""
return [s for s in self._edge.values() if simplex < s]
def _supfaces_id(self, simplex):
"""Returns list of IDs of simplices that contain simplex"""
return [id_ for id_, s in self._edge.items() if simplex < s]
[docs] def add_simplices_from(self, ebunch_to_add, max_order=None, **attr):
r"""Add multiple edges with optional attributes.
Parameters
----------
ebunch_to_add : Iterable
An iterable of simplices. This may be an iterable of iterables (Format 1),
where each element contains the members of the simplex specified as valid
node IDs. Alternatively, each element could also be a tuple in any of the
following formats:
* Format 2: 2-tuple (members, simplex_id), or
* Format 3: 2-tuple (members, attr), or
* Format 4: 3-tuple (members, simplex_id, attr),
where `members` is an iterable of node IDs, `simplex_id` is a hashable to
use as simplex ID, and `attr` is a dict of attributes. Finally,
`ebunch_to_add` may be a dict of the form `{simplex_id: simplex_members}`
(Format 5).
Formats 2 and 3 are unambiguous because `attr` dicts are not hashable, while
`id`s must be. In Formats 2-4, each element of `ebunch_to_add` must have
the same length, i.e. you cannot mix different formats. The iterables
containing simplex members cannot be strings.
max_order : int, optional
Maximal dimension of simplices to add. If None (default), adds all
simplices. If int, and `ebunch_to_add` contains simplices of order >
`max_order`, creates and adds all its subfaces up to `max_order`.
**attr : kwargs, optional
Additional attributes to be assigned to all simplices. Attribues specified
via `ebunch_to_add` take precedence over `attr`.
See Also
--------
add_simplex : add a single simplex
add_weighted_simplices_from : convenient way to add weighted simplices
Notes
-----
Adding the same simplex twice will add it only once. Currently
cannot add empty simplices; the method skips over them.
Examples
--------
>>> import xgi
>>> S = xgi.SimplicialComplex()
When specifying simplices by their members only, numeric simplex IDs will be
assigned automatically.
>>> S.add_simplices_from([[0, 1], [1, 2], [2, 3, 4]])
>>> S.edges.members(dtype=dict) # doctest: +SKIP
{0: frozenset({0, 1}), 1: frozenset({1, 2}), 2: frozenset({2, 3, 4}), 3: frozenset({2, 3}), 4: frozenset({2, 4}), 5: frozenset({3, 4})}
Custom simplex ids can be specified using a dict.
>>> S = xgi.SimplicialComplex()
>>> S.add_simplices_from({'one': [0, 1], 'two': [1, 2], 'three': [2, 3, 4]})
>>> S.edges.members(dtype=dict) # doctest: +SKIP
{'one': frozenset({0, 1}), 'two': frozenset({1, 2}), 'three': frozenset({2, 3, 4}), 0: frozenset({2, 3}), 1: frozenset({2, 4}), 2: frozenset({3, 4})}
You can use the dict format to easily add simplices from another simplicial complex.
>>> S2 = xgi.SimplicialComplex()
>>> S2.add_simplices_from(S.edges.members(dtype=dict))
>>> list(S.edges) == list(S2.edges)
True
Alternatively, simplex ids can be specified using an iterable of 2-tuples.
>>> S = xgi.SimplicialComplex()
>>> S.add_simplices_from([([0, 1], 'one'), ([1, 2], 'two'), ([2, 3, 4], 'three')])
>>> S.edges.members(dtype=dict) # doctest: +SKIP
{'one': frozenset({0, 1}), 'two': frozenset({1, 2}), 'three': frozenset({2, 3, 4}), 0: frozenset({2, 3}), 1: frozenset({2, 4}), 2: frozenset({3, 4})}
Attributes for each simplex may be specified using a 2-tuple for each simplex.
Numeric IDs will be assigned automatically.
>>> S = xgi.SimplicialComplex()
>>> simplices = [
... ([0, 1], {'color': 'red'}),
... ([1, 2], {'age': 30}),
... ([2, 3, 4], {'color': 'blue', 'age': 40}),
... ]
>>> S.add_simplices_from(simplices)
>>> {e: S.edges[e] for e in S.edges}
{0: {'color': 'red'}, 1: {'age': 30}, 2: {'color': 'blue', 'age': 40}, 3: {}, 4: {}, 5: {}}
Attributes and custom IDs may be specified using a 3-tuple for each simplex.
>>> S = xgi.SimplicialComplex()
>>> simplices = [
... ([0, 1], 'one', {'color': 'red'}),
... ([1, 2], 'two', {'age': 30}),
... ([2, 3, 4], 'three', {'color': 'blue', 'age': 40}),
... ]
>>> S.add_simplices_from(simplices)
>>> {e: S.edges[e] for e in S.edges}
{'one': {'color': 'red'}, 'two': {'age': 30}, 'three': {'color': 'blue', 'age': 40}, 0: {}, 1: {}, 2: {}}
"""
# format 5 is the easiest one
if isinstance(ebunch_to_add, dict):
faces = [] # container to store subfaces
for idx, members in ebunch_to_add.items():
# check that it does not exist yet (based on members, not ID)
if not members or self.has_simplex(members):
continue
if idx in self._edge.keys(): # check that uid is not present yet
warn(f"uid {idx} already exists, cannot add simplex {members}.")
continue
if max_order is not None:
if len(members) > max_order + 1:
combos = powerset(members, include_singletons=False)
faces += list(combos)
continue
try:
_ = frozenset(members)
except TypeError as e:
raise XGIError("Invalid ebunch format") from e
self._add_simplex(frozenset(members), idx)
update_uid_counter(self, idx)
# store subfaces
faces += self._subfaces(members)
# add subfaces
faces = set(faces) # get unique subfaces
for members in faces:
# check that it does not exist yet (based on members, not ID)
if not members or self.has_simplex(members):
continue
self._add_face(members)
return
# in formats 1-4 we only know that ebunch_to_add is an iterable, so we iterate
# over it and use the firs element to determine which format we are working with
new_edges = iter(ebunch_to_add)
try:
first_edge = next(new_edges)
except StopIteration:
return
try:
first_elem = list(first_edge)[0]
except TypeError:
first_elem = None
format1, format2, format3, format4 = False, False, False, False
if isinstance(first_elem, Iterable):
if all(isinstance(e, str) for e in first_edge):
format1 = True
elif len(first_edge) == 2 and issubclass(type(first_edge[1]), Hashable):
format2 = True
elif len(first_edge) == 2:
format3 = True
elif len(first_edge) == 3:
format4 = True
else:
format1 = True
if (format1 and isinstance(first_edge, str)) or (
not format1 and isinstance(first_elem, str)
):
raise XGIError("Members cannot be specified as a string")
faces = []
# now we may iterate over the rest
e = first_edge
while True:
if format1:
members, idx, eattr = e, None, {} # uid now set below
elif format2:
members, idx, eattr = e[0], e[1], {}
elif format3:
members, idx, eattr = e[0], None, e[1] # uid now set below
elif format4:
members, idx, eattr = e[0], e[1], e[2]
# check if members is iterable before checking it exists
# to raise meaningful error if not iterable
try:
_ = iter(members)
except TypeError as e:
raise XGIError("Invalid ebunch format") from e
# check that it does not exist yet (based on members, not ID)
if not members or self.has_simplex(members):
try:
e = next(new_edges)
except StopIteration:
break
continue
# needs to go after the check for existence, otherwise
# we're skipping ID numbers when edges already exist
if format1 or format3:
idx = next(self._edge_uid)
if max_order is not None:
if len(members) > max_order + 1:
combos = powerset(members, include_singletons=False)
faces += list(combos) # store faces
try:
e = next(new_edges)
except StopIteration:
break
continue
if idx in self._edge.keys(): # check that uid is not present yet
warn(f"uid {idx} already exists, cannot add simplex {members}.")
try:
e = next(new_edges)
except StopIteration:
break
continue
try:
self._edge[idx] = frozenset(members)
except TypeError as e:
raise XGIError("Invalid ebunch format") from e
for n in members:
if n not in self._node:
self._node[n] = set()
self._node_attr[n] = self._node_attr_dict_factory()
self._node[n].add(idx)
self._edge_attr[idx] = self._edge_attr_dict_factory()
self._edge_attr[idx].update(attr)
self._edge_attr[idx].update(eattr)
update_uid_counter(self, idx)
# store subfaces
faces += self._subfaces(members)
try:
e = next(new_edges)
except StopIteration:
break
# add subfaces
faces = set(faces) # get unique faces
for members in faces:
# check that it does not exist yet (based on members, not ID)
if not members or self.has_simplex(members):
continue
self._add_face(members)
[docs] def close(self):
"""Adds all missing subfaces to the complex.
See Also
--------
add_simplex : add a single simplex
add_weighted_simplices_from : convenient way to add weighted simplices
Notes
-----
Adding the same simplex twice will add it only once. Currently
cannot add empty simplices; the method skips over them.
"""
ebunch_to_close = list(map(list, self.edges.members()))
for simplex in ebunch_to_close:
if isinstance(simplex[-1], dict):
dd = simplex[-1]
simplex = simplex[:-1]
else:
dd = {}
if simplex:
new_faces = self._subfaces(simplex)
self.add_simplices_from(new_faces)
[docs] def add_weighted_simplices_from(
self, ebunch_to_add, max_order=None, weight="weight", **attr
):
"""Add weighted simplices in `ebunch_to_add` with specified weight attr
Parameters
----------
ebunch_to_add : iterable of simplices
Each simplex given in the list or container will be added
to the graph. The simplices must be given as tuples of
the form (node1, node2, ..., noden, weight).
max_order : int, optional
The maximum order simplex to add, by default None.
weight : string, optional
The attribute name for the simplex weights to be added.
By default, "weight".
attr : keyword arguments, optional (default= no attributes)
simplex attributes to add/update for all simplices.
See Also
--------
add_simplex : add a single simplex
add_simplices_from : add multiple simplices
Notes
-----
Adding the same simplex twice will add it only once.
Example
-------
>>> import xgi
>>> S = xgi.SimplicialComplex()
>>> simplices = [(0, 1, 0.3), (0, 2, 0.8)]
>>> S.add_weighted_simplices_from(simplices)
>>> S.edges[0]
{'weight': 0.3}
"""
try:
self.add_simplices_from(
((edge[:-1], {weight: edge[-1]}) for edge in ebunch_to_add),
max_order=max_order,
**attr,
)
except KeyError:
XGIError("Empty or invalid simplices specified.")
def _remove_simplex_id(self, idx):
"""Helper function to remove a simplex with a given id
Parameters
----------
idx : Hashable
edge ID to remove
"""
for node in self.edges.members(idx):
self._node[node].remove(idx)
del self._edge[idx]
del self._edge_attr[idx]
[docs] def remove_simplex_id(self, idx):
"""Remove a simplex with a given id.
This also removes all simplices of which this simplex is face,
to preserve the simplicial complex structure.
Parameters
----------
idx : Hashable
edge ID to remove
Raises
------
XGIError
If no edge has that ID.
See Also
--------
remove_edges_from : remove a collection of edges
"""
try:
# Remove all simplices that contain the given simplex
supfaces_ids = self._supfaces_id(self._edge[idx])
for sup_id in supfaces_ids:
self._remove_simplex_id(sup_id)
# Remove simplex itself
self._remove_simplex_id(idx)
except KeyError as e:
raise XGIError(f"Simplex {idx} is not in the Simplicialcomplex") from e
[docs] def remove_simplex_ids_from(self, ebunch):
"""Remove all simplicies specified in ebunch.
Parameters
----------
ebunch: list or iterable of hashables
Each edge id given in the list or iterable will be removed
from the Simplicialcomplex.
Raises
------
xgi.exception.IDNotFound
If an id in ebunch is not part of the network.
See Also
--------
remove_simplex_id : remove a single simplex by ID.
"""
all_ids = set(self._edge.keys())
for idx in ebunch:
if idx in all_ids and idx not in self._edge.keys():
continue
self.remove_simplex_id(idx)
[docs] def has_simplex(self, simplex):
"""Whether a simplex appears in the simplicial complex.
Parameters
----------
simplex : list or set
An iterable of hashables that specifies an simplex
Returns
-------
bool
Whether or not simplex is as a simplex in the simplicial complex.
Examples
--------
>>> import xgi
>>> S = xgi.SimplicialComplex([[1, 2], [2, 3, 4]])
>>> S.has_simplex([1, 2])
True
>>> S.has_simplex({1, 3})
False
"""
return frozenset(simplex) in self._edge.values()
[docs] def copy(self):
"""A deep copy of the simplicial complex.
A deep copy of the simplicial complex,
including node, edge, and network attributes.
Returns
-------
S : SimplicialComplex
A copy of the simplicial complex.
"""
cp = self.__class__()
nn = self.nodes
cp.add_nodes_from((n, deepcopy(attr)) for n, attr in nn.items())
ee = self.edges
cp.add_simplices_from(
(e, idx, deepcopy(self.edges[idx]))
for idx, e in ee.members(dtype=dict).items()
)
cp._net_attr = deepcopy(self._net_attr)
cp._edge_uid = copy(self._edge_uid)
return cp
[docs] def cleanup(self, isolates=False, connected=True, relabel=True, in_place=True):
if in_place:
_S = self
else:
_S = self.copy()
if not isolates:
_S.remove_nodes_from(_S.nodes.isolates())
if connected:
from ..algorithms import largest_connected_hypergraph
largest_connected_hypergraph(_S, in_place=True)
if relabel:
from ..utils import convert_labels_to_integers
convert_labels_to_integers(_S, in_place=True)
return _S
[docs] def freeze(self):
"""Method for freezing a simplicial complex
which prevents it from being modified
See Also
--------
frozen : Method that raises an error when a
user tries to modify the simplicial complex
is_frozen : Check whether a simplicial complex is frozen
Examples
--------
>>> import xgi
>>> edges = [[1, 2], [2, 3, 4]]
>>> S = xgi.SimplicialComplex(edges)
>>> S.freeze()
>>> S.add_node(5)
Traceback (most recent call last):
xgi.exception.XGIError: Frozen higher-order network can't be modified
"""
self.add_node = frozen
self.add_nodes_from = frozen
self.remove_node = frozen
self.remove_nodes_from = frozen
self.add_simplex = frozen
self.add_simplices_from = frozen
self.add_weighted_simplices_from = frozen
self.remove_simplex_id = frozen
self.remove_simplex_ids_from = frozen
self.clear = frozen
self.frozen = True
@property
def is_frozen(self):
"""Checks whether a simplicial complex is frozen
Returns
-------
bool
True if simplicial complex is frozen, false if not.
See Also
--------
freeze : A method to prevent a simplicial complex from being modified.
Examples
--------
>>> import xgi
>>> edges = [[1, 2], [2, 3, 4]]
>>> S = xgi.SimplicialComplex(edges)
>>> S.freeze()
>>> S.is_frozen
True
"""
try:
return self.frozen
except AttributeError:
return False
