# Source code for strawberryfields.apps.subgraph

```
# Copyright 2019 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
r"""
Tools for users to find dense subgraphs of different sizes.
The search heuristic in this module works by resizing a collection of starting subgraphs and
keeping track of the densest identified. The starting subgraphs can be selected by sampling from
GBS, resulting in candidates that are likely to be dense :cite:`arrazola2018using`.
.. seealso::
:ref:`apps-subgraph-tutorial`
Algorithm
^^^^^^^^^
The heuristic algorithm provided proceeds as follows. Each starting subgraph :math:`s` is resized
to a range of sizes :math:`\{k\}_{k_{\min}}^{k_{\max}}`, resulting in the subgraphs
:math:`s_{k}`. For a given size :math:`k`, a collection of the densest :math:`n` subgraphs
identified is recorded, meaning that :math:`s_{k}` is added to the collection only if it has
sufficient density.
This algorithm returns a dictionary over the range of sizes specified, with each value being the
collection of densest-:math:`k` subgraphs. This collection is a list of tuple pairs specifying
the subgraph density and nodes.
Subgraph resizing
^^^^^^^^^^^^^^^^^
The key element of the :func:`search` algorithm is the resizing of each subgraph, allowing a
range of subgraph sizes to be tracked. Resizing proceeds in the :func:`resize` function by greedily
adding or removing nodes to a subgraph one-at-a-time. Node selection is carried out by picking
the node with the greatest or least degree with respect to the subgraph. This function returns a
dictionary over the range of sizes specified, with each value being the corresponding resized
subgraph.
Whenever there are multiple candidate nodes with the same degree, there must be a choice of
which node to add or remove. The supported choices are:
- Select among candidate nodes uniformly at random;
- Select the candidate node with the greatest node weight, settling remaining ties uniformly at
random.
"""
from typing import Tuple, Union
import networkx as nx
import numpy as np
[docs]def search(
subgraphs: list,
graph: nx.Graph,
min_size: int,
max_size: int,
max_count: int = 10,
node_select: Union[str, np.ndarray, list] = "uniform",
) -> dict:
"""Search for dense subgraphs within an input size range.
For each subgraph from ``subgraphs``, this function resizes using :func:`resize` to the input
range specified by ``min_size`` and ``max_size``, resulting in a range of differently sized
subgraphs. This function loops over all elements of ``subgraphs`` and keeps track of the
``max_count`` number of densest subgraphs identified for each size.
In both growth and shrink phases of :func:`resize`, there may be multiple candidate nodes with
equal degree to add to or remove from the subgraph. The method of selecting the node is
specified by the ``node_select`` argument, which can be either:
- ``"uniform"`` (default): choose a node from the candidates uniformly at random;
- A list or array: specifying the node weights of the graph, resulting in choosing the node
from the candidates with the highest weight (when growing) and lowest weight (when shrinking),
settling remaining ties by uniform random choice.
**Example usage:**
>>> s = data.Planted()
>>> g = nx.Graph(s.adj)
>>> s = sample.postselect(s, 16, 30)
>>> s = sample.to_subgraphs(s, g)
>>> search(s, g, 8, 9, max_count=3)
{9: [(0.9722222222222222, [21, 22, 23, 24, 25, 26, 27, 28, 29]),
(0.9722222222222222, [20, 21, 22, 24, 25, 26, 27, 28, 29]),
(0.9444444444444444, [20, 21, 22, 23, 24, 25, 26, 27, 29])],
8: [(1.0, [21, 22, 24, 25, 26, 27, 28, 29]),
(1.0, [21, 22, 23, 24, 25, 26, 27, 28]),
(1.0, [20, 21, 22, 24, 25, 26, 27, 29])]}
Args:
subgraphs (list[list[int]]): a list of subgraphs specified by their nodes
graph (nx.Graph): the input graph
min_size (int): minimum size to search for dense subgraphs
max_size (int): maximum size to search for dense subgraphs
max_count (int): maximum number of densest subgraphs to keep track of for each size
node_select (str, list or array): method of settling ties when more than one node of
equal degree can be added/removed. Can be ``"uniform"`` (default), or a NumPy array or
list containing node weights.
Returns:
dict[int, list[tuple[float, list[int]]]]: a dictionary of different sizes, each containing a
list of densest subgraphs reported as a tuple of subgraph density and subgraph nodes,
sorted in non-increasing order of density
"""
dense = {}
for s in subgraphs:
r = resize(s, graph, min_size, max_size, node_select)
for size, subgraph in r.items():
r[size] = (nx.density(graph.subgraph(subgraph)), subgraph)
_update_dict(dense, r, max_count)
return dense
def _update_dict(d: dict, d_new: dict, max_count: int) -> None:
"""Updates dictionary ``d`` with subgraph tuples contained in ``d_new``.
Subgraph tuples are a pair of values: a float specifying the subgraph density and a list of
integers specifying the subgraph nodes. Both ``d`` and ``d_new`` are dictionaries over
different subgraph sizes. The values of ``d`` are lists of subgraph tuples containing the top
densest subgraphs for a given size, with maximum length ``max_count``. The values of
``d_new`` are candidate subgraph tuples that can be the result of resizing an input subgraph
over a range using :func:`resize`. We want to add these candidates to the list of subgraph
tuples in ``d`` to build up our collection of dense subgraphs.
Args:
d (dict[int, list[tuple[float, list[int]]]]): dictionary of subgraph sizes and
corresponding list of subgraph tuples
d_new (dict[int, tuple[float, list[int]]]): dictionary of subgraph sizes and corresponding
subgraph tuples that are candidates to be added to the list
max_count (int): the maximum length of every subgraph tuple list
Returns:
None: this function modifies the dictionary ``d`` in place
"""
for size, t in d_new.items():
l = d.setdefault(size, [t])
_update_subgraphs_list(l, t, max_count)
def _update_subgraphs_list(l: list, t: tuple, max_count: int) -> None:
"""Updates list of top subgraphs with a candidate.
Here, the list ``l`` to be updated is a list of tuples with each tuple being a pair of
values: a float specifying the subgraph density and a list of integers specifying the
subgraph nodes. For example, ``l`` may be:
``[(0.8, [0, 5, 9, 10]), (0.5, [1, 2, 5, 6]), (0.3, [0, 4, 6, 9])]``
We want to update ``l`` with a candidate tuple ``t``, which should be a pair specifying a
subgraph density and corresponding subgraph nodes. For example, we might want to add:
``(0.4, [1, 4, 9, 10])``
This function checks:
- if ``t`` is already an element of ``l``, do nothing (i.e., so that ``l`` never has
repetitions)
- if ``len(l) < max_count``, add ``t``
- otherwise, if the density of ``t`` exceeds the minimum density of ``l`` , add ``t`` and
remove the element with the minimum density
- otherwise, if the density of ``t`` equals the minimum density of ``l``, flip a coin and
randomly swap in ``t`` with the minimum element of ``l``.
The list ``l`` is also sorted so that its first element is the subgraph with the highest
density.
Args:
l (list[tuple[float, list[int]]]): the list of subgraph tuples to be updated
t (tuple[float, list[int]): the candidate subgraph tuple
max_count (int): the maximum length of ``l``
Returns:
None: this function modifies ``l`` in place
"""
t = (t[0], sorted(set(t[1])))
for _d, s in l:
if t[1] == s:
return
if len(l) < max_count:
l.append(t)
l.sort(reverse=True)
return
l_min = l[-1][0]
if t[0] > l_min:
l.append(t)
l.sort(reverse=True)
del l[-1]
elif t[0] == l_min:
if np.random.choice(2):
del l[-1]
l.append(t)
l.sort(reverse=True)
return
[docs]def resize(
subgraph: list,
graph: nx.Graph,
min_size: int,
max_size: int,
node_select: Union[str, np.ndarray, list] = "uniform",
) -> dict:
"""Resize a subgraph to a range of input sizes.
This function uses a greedy approach to iteratively add or remove nodes one at a time to an
input subgraph to reach the range of sizes specified by ``min_size`` and ``max_size``.
When growth is required, the algorithm examines all nodes from the remainder of the graph as
candidates and adds the single node with the highest degree relative to the rest of the
subgraph. This results in a graph that is one node larger, and if growth is still required,
the algorithm performs the procedure again.
When shrinking is required, the algorithm examines all nodes from within the subgraph as
candidates and removes the single node with lowest degree relative to the subgraph.
In both growth and shrink phases, there may be multiple candidate nodes with equal degree to
add to or remove from the subgraph. The method of selecting the node is specified by the
``node_select`` argument, which can be either:
- ``"uniform"`` (default): choose a node from the candidates uniformly at random;
- A list or array: specifying the node weights of the graph, resulting in choosing the node
from the candidates with the highest weight (when growing) and lowest weight (when shrinking),
settling remaining ties by uniform random choice.
**Example usage:**
>>> s = data.Planted()
>>> g = nx.Graph(s.adj)
>>> s = [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]
>>> resize(s, g, 8, 12)
{10: [20, 21, 22, 23, 24, 25, 26, 27, 28, 29],
11: [11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29],
12: [0, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29],
9: [20, 21, 22, 24, 25, 26, 27, 28, 29],
8: [20, 21, 22, 24, 25, 26, 27, 29]}
Args:
subgraph (list[int]): a subgraph specified by a list of nodes
graph (nx.Graph): the input graph
min_size (int): minimum size for subgraph to be resized to
max_size (int): maximum size for subgraph to be resized to
node_select (str, list or array): method of settling ties when more than one node of
equal degree can be added/removed. Can be ``"uniform"`` (default), or a NumPy array or
list containing node weights.
Returns:
dict[int, list[int]]: a dictionary of different sizes with corresponding subgraph
"""
nodes = graph.nodes()
subgraph = set(subgraph)
node_select, w = _validate_inputs(subgraph, graph, min_size, max_size, node_select)
starting_size = len(subgraph)
if min_size <= starting_size <= max_size:
resized = {starting_size: sorted(subgraph)}
else:
resized = {}
if max_size > starting_size:
grow_subgraph = graph.subgraph(subgraph).copy()
while grow_subgraph.order() < max_size:
grow_nodes = grow_subgraph.nodes()
complement_nodes = nodes - grow_nodes
degrees = np.array(
[(c, graph.subgraph(list(grow_nodes) + [c]).degree()[c]) for c in complement_nodes]
)
degrees_max = np.argwhere(degrees[:, 1] == degrees[:, 1].max()).flatten()
if node_select == "uniform":
to_add_index = np.random.choice(degrees_max)
elif node_select == "weight":
weights = np.array([w[degrees[n][0]] for n in degrees_max])
to_add_index = np.random.choice(np.where(weights == weights.max())[0])
to_add = degrees[to_add_index][0]
grow_subgraph.add_node(to_add)
new_size = grow_subgraph.order()
if min_size <= new_size <= max_size:
resized[new_size] = sorted(grow_subgraph.nodes())
if min_size < starting_size:
shrink_subgraph = graph.subgraph(subgraph).copy()
while shrink_subgraph.order() > min_size:
degrees = np.array(shrink_subgraph.degree)
degrees_min = np.argwhere(degrees[:, 1] == degrees[:, 1].min()).flatten()
if node_select == "uniform":
to_remove_index = np.random.choice(degrees_min)
elif node_select == "weight":
weights = np.array([w[degrees[n][0]] for n in degrees_min])
to_remove_index = np.random.choice(np.where(weights == weights.min())[0])
to_remove = degrees[to_remove_index][0]
shrink_subgraph.remove_node(to_remove)
new_size = shrink_subgraph.order()
if min_size <= new_size <= max_size:
resized[new_size] = sorted(shrink_subgraph.nodes())
return resized
def _validate_inputs(
subgraph: set,
graph: nx.Graph,
min_size: int,
max_size: int,
node_select: Union[str, np.ndarray, list] = "uniform",
) -> Tuple:
"""Validates input for the ``resize`` function.
This function checks:
- if ``subgraph`` is a valid subgraph of ``graph``;
- if ``min_size`` and ``max_size`` are sensible numbers;
- if ``node_select`` is either ``"uniform"`` or a NumPy array or list;
- if, when ``node_select`` is a NumPy array or list, that it is the correct size and that
``node_select`` is changed to ``"weight"``.
This function returns the updated ``node_select`` and a dictionary mapping nodes to their
corresponding weights (weights default to unity if not specified).
Args:
subgraph (list[int]): a subgraph specified by a list of nodes
graph (nx.Graph): the input graph
min_size (int): minimum size for subgraph to be resized to
max_size (int): maximum size for subgraph to be resized to
node_select (str, list or array): method of settling ties when more than one node of
equal degree can be added/removed. Can be ``"uniform"`` (default), or a NumPy array or
list containing node weights.
Returns:
tuple[str, dict]: the updated ``node_select`` and a dictionary of node weights
"""
if not subgraph.issubset(graph.nodes()):
raise ValueError("Input is not a valid subgraph")
if min_size < 1:
raise ValueError("min_size must be at least 1")
if max_size >= len(graph.nodes()):
raise ValueError("max_size must be less than number of nodes in graph")
if max_size < min_size:
raise ValueError("max_size must not be less than min_size")
if isinstance(node_select, (list, np.ndarray)):
if len(node_select) != graph.number_of_nodes():
raise ValueError("Number of node weights must match number of nodes")
w = {n: node_select[i] for i, n in enumerate(graph.nodes)}
node_select = "weight"
else:
w = {n: 1 for i, n in enumerate(graph.nodes)}
if node_select != "uniform":
raise ValueError("Node selection method not recognized")
return node_select, w
```

_modules/strawberryfields/apps/subgraph

Download Python script

Download Notebook

View on GitHub