406 lines
14 KiB
Python
406 lines
14 KiB
Python
|
"""Swap edges in a graph.
|
||
|
|
"""
|
||
|
|
|
||
|
|
import math
|
||
|
|
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.utils import py_random_state
|
||
|
|
|
||
|
|
__all__ = ["double_edge_swap", "connected_double_edge_swap", "directed_edge_swap"]
|
||
|
|
|
||
|
|
|
||
|
|
@nx.utils.not_implemented_for("undirected")
|
||
|
|
@py_random_state(3)
|
||
|
|
@nx._dispatch
|
||
|
|
def directed_edge_swap(G, *, nswap=1, max_tries=100, seed=None):
|
||
|
|
"""Swap three edges in a directed graph while keeping the node degrees fixed.
|
||
|
|
|
||
|
|
A directed edge swap swaps three edges such that a -> b -> c -> d becomes
|
||
|
|
a -> c -> b -> d. This pattern of swapping allows all possible states with the
|
||
|
|
same in- and out-degree distribution in a directed graph to be reached.
|
||
|
|
|
||
|
|
If the swap would create parallel edges (e.g. if a -> c already existed in the
|
||
|
|
previous example), another attempt is made to find a suitable trio of edges.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : DiGraph
|
||
|
|
A directed graph
|
||
|
|
|
||
|
|
nswap : integer (optional, default=1)
|
||
|
|
Number of three-edge (directed) swaps to perform
|
||
|
|
|
||
|
|
max_tries : integer (optional, default=100)
|
||
|
|
Maximum number of attempts to swap edges
|
||
|
|
|
||
|
|
seed : integer, random_state, or None (default)
|
||
|
|
Indicator of random number generation state.
|
||
|
|
See :ref:`Randomness<randomness>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
G : DiGraph
|
||
|
|
The graph after the edges are swapped.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
NetworkXError
|
||
|
|
If `G` is not directed, or
|
||
|
|
If nswap > max_tries, or
|
||
|
|
If there are fewer than 4 nodes or 3 edges in `G`.
|
||
|
|
NetworkXAlgorithmError
|
||
|
|
If the number of swap attempts exceeds `max_tries` before `nswap` swaps are made
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Does not enforce any connectivity constraints.
|
||
|
|
|
||
|
|
The graph G is modified in place.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Erdős, Péter L., et al. “A Simple Havel-Hakimi Type Algorithm to Realize
|
||
|
|
Graphical Degree Sequences of Directed Graphs.” ArXiv:0905.4913 [Math],
|
||
|
|
Jan. 2010. https://doi.org/10.48550/arXiv.0905.4913.
|
||
|
|
Published 2010 in Elec. J. Combinatorics (17(1)). R66.
|
||
|
|
http://www.combinatorics.org/Volume_17/PDF/v17i1r66.pdf
|
||
|
|
.. [2] “Combinatorics - Reaching All Possible Simple Directed Graphs with a given
|
||
|
|
Degree Sequence with 2-Edge Swaps.” Mathematics Stack Exchange,
|
||
|
|
https://math.stackexchange.com/questions/22272/. Accessed 30 May 2022.
|
||
|
|
"""
|
||
|
|
if nswap > max_tries:
|
||
|
|
raise nx.NetworkXError("Number of swaps > number of tries allowed.")
|
||
|
|
if len(G) < 4:
|
||
|
|
raise nx.NetworkXError("DiGraph has fewer than four nodes.")
|
||
|
|
if len(G.edges) < 3:
|
||
|
|
raise nx.NetworkXError("DiGraph has fewer than 3 edges")
|
||
|
|
|
||
|
|
# Instead of choosing uniformly at random from a generated edge list,
|
||
|
|
# this algorithm chooses nonuniformly from the set of nodes with
|
||
|
|
# probability weighted by degree.
|
||
|
|
tries = 0
|
||
|
|
swapcount = 0
|
||
|
|
keys, degrees = zip(*G.degree()) # keys, degree
|
||
|
|
cdf = nx.utils.cumulative_distribution(degrees) # cdf of degree
|
||
|
|
discrete_sequence = nx.utils.discrete_sequence
|
||
|
|
|
||
|
|
while swapcount < nswap:
|
||
|
|
# choose source node index from discrete distribution
|
||
|
|
start_index = discrete_sequence(1, cdistribution=cdf, seed=seed)[0]
|
||
|
|
start = keys[start_index]
|
||
|
|
tries += 1
|
||
|
|
|
||
|
|
if tries > max_tries:
|
||
|
|
msg = f"Maximum number of swap attempts ({tries}) exceeded before desired swaps achieved ({nswap})."
|
||
|
|
raise nx.NetworkXAlgorithmError(msg)
|
||
|
|
|
||
|
|
# If the given node doesn't have any out edges, then there isn't anything to swap
|
||
|
|
if G.out_degree(start) == 0:
|
||
|
|
continue
|
||
|
|
second = seed.choice(list(G.succ[start]))
|
||
|
|
if start == second:
|
||
|
|
continue
|
||
|
|
|
||
|
|
if G.out_degree(second) == 0:
|
||
|
|
continue
|
||
|
|
third = seed.choice(list(G.succ[second]))
|
||
|
|
if second == third:
|
||
|
|
continue
|
||
|
|
|
||
|
|
if G.out_degree(third) == 0:
|
||
|
|
continue
|
||
|
|
fourth = seed.choice(list(G.succ[third]))
|
||
|
|
if third == fourth:
|
||
|
|
continue
|
||
|
|
|
||
|
|
if (
|
||
|
|
third not in G.succ[start]
|
||
|
|
and fourth not in G.succ[second]
|
||
|
|
and second not in G.succ[third]
|
||
|
|
):
|
||
|
|
# Swap nodes
|
||
|
|
G.add_edge(start, third)
|
||
|
|
G.add_edge(third, second)
|
||
|
|
G.add_edge(second, fourth)
|
||
|
|
G.remove_edge(start, second)
|
||
|
|
G.remove_edge(second, third)
|
||
|
|
G.remove_edge(third, fourth)
|
||
|
|
swapcount += 1
|
||
|
|
|
||
|
|
return G
|
||
|
|
|
||
|
|
|
||
|
|
@py_random_state(3)
|
||
|
|
@nx._dispatch
|
||
|
|
def double_edge_swap(G, nswap=1, max_tries=100, seed=None):
|
||
|
|
"""Swap two edges in the graph while keeping the node degrees fixed.
|
||
|
|
|
||
|
|
A double-edge swap removes two randomly chosen edges u-v and x-y
|
||
|
|
and creates the new edges u-x and v-y::
|
||
|
|
|
||
|
|
u--v u v
|
||
|
|
becomes | |
|
||
|
|
x--y x y
|
||
|
|
|
||
|
|
If either the edge u-x or v-y already exist no swap is performed
|
||
|
|
and another attempt is made to find a suitable edge pair.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : graph
|
||
|
|
An undirected graph
|
||
|
|
|
||
|
|
nswap : integer (optional, default=1)
|
||
|
|
Number of double-edge swaps to perform
|
||
|
|
|
||
|
|
max_tries : integer (optional)
|
||
|
|
Maximum number of attempts to swap edges
|
||
|
|
|
||
|
|
seed : integer, random_state, or None (default)
|
||
|
|
Indicator of random number generation state.
|
||
|
|
See :ref:`Randomness<randomness>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
G : graph
|
||
|
|
The graph after double edge swaps.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
NetworkXError
|
||
|
|
If `G` is directed, or
|
||
|
|
If `nswap` > `max_tries`, or
|
||
|
|
If there are fewer than 4 nodes or 2 edges in `G`.
|
||
|
|
NetworkXAlgorithmError
|
||
|
|
If the number of swap attempts exceeds `max_tries` before `nswap` swaps are made
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Does not enforce any connectivity constraints.
|
||
|
|
|
||
|
|
The graph G is modified in place.
|
||
|
|
"""
|
||
|
|
if G.is_directed():
|
||
|
|
raise nx.NetworkXError(
|
||
|
|
"double_edge_swap() not defined for directed graphs. Use directed_edge_swap instead."
|
||
|
|
)
|
||
|
|
if nswap > max_tries:
|
||
|
|
raise nx.NetworkXError("Number of swaps > number of tries allowed.")
|
||
|
|
if len(G) < 4:
|
||
|
|
raise nx.NetworkXError("Graph has fewer than four nodes.")
|
||
|
|
if len(G.edges) < 2:
|
||
|
|
raise nx.NetworkXError("Graph has fewer than 2 edges")
|
||
|
|
# Instead of choosing uniformly at random from a generated edge list,
|
||
|
|
# this algorithm chooses nonuniformly from the set of nodes with
|
||
|
|
# probability weighted by degree.
|
||
|
|
n = 0
|
||
|
|
swapcount = 0
|
||
|
|
keys, degrees = zip(*G.degree()) # keys, degree
|
||
|
|
cdf = nx.utils.cumulative_distribution(degrees) # cdf of degree
|
||
|
|
discrete_sequence = nx.utils.discrete_sequence
|
||
|
|
while swapcount < nswap:
|
||
|
|
# if random.random() < 0.5: continue # trick to avoid periodicities?
|
||
|
|
# pick two random edges without creating edge list
|
||
|
|
# choose source node indices from discrete distribution
|
||
|
|
(ui, xi) = discrete_sequence(2, cdistribution=cdf, seed=seed)
|
||
|
|
if ui == xi:
|
||
|
|
continue # same source, skip
|
||
|
|
u = keys[ui] # convert index to label
|
||
|
|
x = keys[xi]
|
||
|
|
# choose target uniformly from neighbors
|
||
|
|
v = seed.choice(list(G[u]))
|
||
|
|
y = seed.choice(list(G[x]))
|
||
|
|
if v == y:
|
||
|
|
continue # same target, skip
|
||
|
|
if (x not in G[u]) and (y not in G[v]): # don't create parallel edges
|
||
|
|
G.add_edge(u, x)
|
||
|
|
G.add_edge(v, y)
|
||
|
|
G.remove_edge(u, v)
|
||
|
|
G.remove_edge(x, y)
|
||
|
|
swapcount += 1
|
||
|
|
if n >= max_tries:
|
||
|
|
e = (
|
||
|
|
f"Maximum number of swap attempts ({n}) exceeded "
|
||
|
|
f"before desired swaps achieved ({nswap})."
|
||
|
|
)
|
||
|
|
raise nx.NetworkXAlgorithmError(e)
|
||
|
|
n += 1
|
||
|
|
return G
|
||
|
|
|
||
|
|
|
||
|
|
@py_random_state(3)
|
||
|
|
@nx._dispatch
|
||
|
|
def connected_double_edge_swap(G, nswap=1, _window_threshold=3, seed=None):
|
||
|
|
"""Attempts the specified number of double-edge swaps in the graph `G`.
|
||
|
|
|
||
|
|
A double-edge swap removes two randomly chosen edges `(u, v)` and `(x,
|
||
|
|
y)` and creates the new edges `(u, x)` and `(v, y)`::
|
||
|
|
|
||
|
|
u--v u v
|
||
|
|
becomes | |
|
||
|
|
x--y x y
|
||
|
|
|
||
|
|
If either `(u, x)` or `(v, y)` already exist, then no swap is performed
|
||
|
|
so the actual number of swapped edges is always *at most* `nswap`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : graph
|
||
|
|
An undirected graph
|
||
|
|
|
||
|
|
nswap : integer (optional, default=1)
|
||
|
|
Number of double-edge swaps to perform
|
||
|
|
|
||
|
|
_window_threshold : integer
|
||
|
|
|
||
|
|
The window size below which connectedness of the graph will be checked
|
||
|
|
after each swap.
|
||
|
|
|
||
|
|
The "window" in this function is a dynamically updated integer that
|
||
|
|
represents the number of swap attempts to make before checking if the
|
||
|
|
graph remains connected. It is an optimization used to decrease the
|
||
|
|
running time of the algorithm in exchange for increased complexity of
|
||
|
|
implementation.
|
||
|
|
|
||
|
|
If the window size is below this threshold, then the algorithm checks
|
||
|
|
after each swap if the graph remains connected by checking if there is a
|
||
|
|
path joining the two nodes whose edge was just removed. If the window
|
||
|
|
size is above this threshold, then the algorithm performs do all the
|
||
|
|
swaps in the window and only then check if the graph is still connected.
|
||
|
|
|
||
|
|
seed : integer, random_state, or None (default)
|
||
|
|
Indicator of random number generation state.
|
||
|
|
See :ref:`Randomness<randomness>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
int
|
||
|
|
The number of successful swaps
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
|
||
|
|
NetworkXError
|
||
|
|
|
||
|
|
If the input graph is not connected, or if the graph has fewer than four
|
||
|
|
nodes.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
The initial graph `G` must be connected, and the resulting graph is
|
||
|
|
connected. The graph `G` is modified in place.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] C. Gkantsidis and M. Mihail and E. Zegura,
|
||
|
|
The Markov chain simulation method for generating connected
|
||
|
|
power law random graphs, 2003.
|
||
|
|
http://citeseer.ist.psu.edu/gkantsidis03markov.html
|
||
|
|
"""
|
||
|
|
if not nx.is_connected(G):
|
||
|
|
raise nx.NetworkXError("Graph not connected")
|
||
|
|
if len(G) < 4:
|
||
|
|
raise nx.NetworkXError("Graph has fewer than four nodes.")
|
||
|
|
n = 0
|
||
|
|
swapcount = 0
|
||
|
|
deg = G.degree()
|
||
|
|
# Label key for nodes
|
||
|
|
dk = [n for n, d in G.degree()]
|
||
|
|
cdf = nx.utils.cumulative_distribution([d for n, d in G.degree()])
|
||
|
|
discrete_sequence = nx.utils.discrete_sequence
|
||
|
|
window = 1
|
||
|
|
while n < nswap:
|
||
|
|
wcount = 0
|
||
|
|
swapped = []
|
||
|
|
# If the window is small, we just check each time whether the graph is
|
||
|
|
# connected by checking if the nodes that were just separated are still
|
||
|
|
# connected.
|
||
|
|
if window < _window_threshold:
|
||
|
|
# This Boolean keeps track of whether there was a failure or not.
|
||
|
|
fail = False
|
||
|
|
while wcount < window and n < nswap:
|
||
|
|
# Pick two random edges without creating the edge list. Choose
|
||
|
|
# source nodes from the discrete degree distribution.
|
||
|
|
(ui, xi) = discrete_sequence(2, cdistribution=cdf, seed=seed)
|
||
|
|
# If the source nodes are the same, skip this pair.
|
||
|
|
if ui == xi:
|
||
|
|
continue
|
||
|
|
# Convert an index to a node label.
|
||
|
|
u = dk[ui]
|
||
|
|
x = dk[xi]
|
||
|
|
# Choose targets uniformly from neighbors.
|
||
|
|
v = seed.choice(list(G.neighbors(u)))
|
||
|
|
y = seed.choice(list(G.neighbors(x)))
|
||
|
|
# If the target nodes are the same, skip this pair.
|
||
|
|
if v == y:
|
||
|
|
continue
|
||
|
|
if x not in G[u] and y not in G[v]:
|
||
|
|
G.remove_edge(u, v)
|
||
|
|
G.remove_edge(x, y)
|
||
|
|
G.add_edge(u, x)
|
||
|
|
G.add_edge(v, y)
|
||
|
|
swapped.append((u, v, x, y))
|
||
|
|
swapcount += 1
|
||
|
|
n += 1
|
||
|
|
# If G remains connected...
|
||
|
|
if nx.has_path(G, u, v):
|
||
|
|
wcount += 1
|
||
|
|
# Otherwise, undo the changes.
|
||
|
|
else:
|
||
|
|
G.add_edge(u, v)
|
||
|
|
G.add_edge(x, y)
|
||
|
|
G.remove_edge(u, x)
|
||
|
|
G.remove_edge(v, y)
|
||
|
|
swapcount -= 1
|
||
|
|
fail = True
|
||
|
|
# If one of the swaps failed, reduce the window size.
|
||
|
|
if fail:
|
||
|
|
window = math.ceil(window / 2)
|
||
|
|
else:
|
||
|
|
window += 1
|
||
|
|
# If the window is large, then there is a good chance that a bunch of
|
||
|
|
# swaps will work. It's quicker to do all those swaps first and then
|
||
|
|
# check if the graph remains connected.
|
||
|
|
else:
|
||
|
|
while wcount < window and n < nswap:
|
||
|
|
# Pick two random edges without creating the edge list. Choose
|
||
|
|
# source nodes from the discrete degree distribution.
|
||
|
|
(ui, xi) = discrete_sequence(2, cdistribution=cdf, seed=seed)
|
||
|
|
# If the source nodes are the same, skip this pair.
|
||
|
|
if ui == xi:
|
||
|
|
continue
|
||
|
|
# Convert an index to a node label.
|
||
|
|
u = dk[ui]
|
||
|
|
x = dk[xi]
|
||
|
|
# Choose targets uniformly from neighbors.
|
||
|
|
v = seed.choice(list(G.neighbors(u)))
|
||
|
|
y = seed.choice(list(G.neighbors(x)))
|
||
|
|
# If the target nodes are the same, skip this pair.
|
||
|
|
if v == y:
|
||
|
|
continue
|
||
|
|
if x not in G[u] and y not in G[v]:
|
||
|
|
G.remove_edge(u, v)
|
||
|
|
G.remove_edge(x, y)
|
||
|
|
G.add_edge(u, x)
|
||
|
|
G.add_edge(v, y)
|
||
|
|
swapped.append((u, v, x, y))
|
||
|
|
swapcount += 1
|
||
|
|
n += 1
|
||
|
|
wcount += 1
|
||
|
|
# If the graph remains connected, increase the window size.
|
||
|
|
if nx.is_connected(G):
|
||
|
|
window += 1
|
||
|
|
# Otherwise, undo the changes from the previous window and decrease
|
||
|
|
# the window size.
|
||
|
|
else:
|
||
|
|
while swapped:
|
||
|
|
(u, v, x, y) = swapped.pop()
|
||
|
|
G.add_edge(u, v)
|
||
|
|
G.add_edge(x, y)
|
||
|
|
G.remove_edge(u, x)
|
||
|
|
G.remove_edge(v, y)
|
||
|
|
swapcount -= 1
|
||
|
|
window = math.ceil(window / 2)
|
||
|
|
return swapcount
|