"""
==========================
Bipartite Graph Algorithms
==========================
"""
import networkx as nx
from networkx.algorithms.components import connected_components
__all__ = [
"is_bipartite",
"is_bipartite_node_set",
"color",
"sets",
"density",
"degrees",
]
[docs]def color(G):
"""Returns a two-coloring of the graph.
Raises an exception if the graph is not bipartite.
Parameters
----------
G : NetworkX graph
Returns
-------
color : dictionary
A dictionary keyed by node with a 1 or 0 as data for each node color.
Raises
------
NetworkXError
If the graph is not two-colorable.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> c = bipartite.color(G)
>>> print(c)
{0: 1, 1: 0, 2: 1, 3: 0}
You can use this to set a node attribute indicating the biparite set:
>>> nx.set_node_attributes(G, c, "bipartite")
>>> print(G.nodes[0]["bipartite"])
1
>>> print(G.nodes[1]["bipartite"])
0
"""
if G.is_directed():
import itertools
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])
else:
neighbors = G.neighbors
color = {}
for n in G: # handle disconnected graphs
if n in color or len(G[n]) == 0: # skip isolates
continue
queue = [n]
color[n] = 1 # nodes seen with color (1 or 0)
while queue:
v = queue.pop()
c = 1 - color[v] # opposite color of node v
for w in neighbors(v):
if w in color:
if color[w] == color[v]:
raise nx.NetworkXError("Graph is not bipartite.")
else:
color[w] = c
queue.append(w)
# color isolates with 0
color.update(dict.fromkeys(nx.isolates(G), 0))
return color
[docs]def is_bipartite(G):
""" Returns True if graph G is bipartite, False if not.
Parameters
----------
G : NetworkX graph
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> print(bipartite.is_bipartite(G))
True
See Also
--------
color, is_bipartite_node_set
"""
try:
color(G)
return True
except nx.NetworkXError:
return False
[docs]def is_bipartite_node_set(G, nodes):
"""Returns True if nodes and G/nodes are a bipartition of G.
Parameters
----------
G : NetworkX graph
nodes: list or container
Check if nodes are a one of a bipartite set.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> X = set([1, 3])
>>> bipartite.is_bipartite_node_set(G, X)
True
Notes
-----
For connected graphs the bipartite sets are unique. This function handles
disconnected graphs.
"""
S = set(nodes)
for CC in (G.subgraph(c).copy() for c in connected_components(G)):
X, Y = sets(CC)
if not (
(X.issubset(S) and Y.isdisjoint(S)) or (Y.issubset(S) and X.isdisjoint(S))
):
return False
return True
[docs]def sets(G, top_nodes=None):
"""Returns bipartite node sets of graph G.
Raises an exception if the graph is not bipartite or if the input
graph is disconnected and thus more than one valid solution exists.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
Parameters
----------
G : NetworkX graph
top_nodes : container, optional
Container with all nodes in one bipartite node set. If not supplied
it will be computed. But if more than one solution exists an exception
will be raised.
Returns
-------
X : set
Nodes from one side of the bipartite graph.
Y : set
Nodes from the other side.
Raises
------
AmbiguousSolution
Raised if the input bipartite graph is disconnected and no container
with all nodes in one bipartite set is provided. When determining
the nodes in each bipartite set more than one valid solution is
possible if the input graph is disconnected.
NetworkXError
Raised if the input graph is not bipartite.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> X, Y = bipartite.sets(G)
>>> list(X)
[0, 2]
>>> list(Y)
[1, 3]
See Also
--------
color
"""
if G.is_directed():
is_connected = nx.is_weakly_connected
else:
is_connected = nx.is_connected
if top_nodes is not None:
X = set(top_nodes)
Y = set(G) - X
else:
if not is_connected(G):
msg = "Disconnected graph: Ambiguous solution for bipartite sets."
raise nx.AmbiguousSolution(msg)
c = color(G)
X = {n for n, is_top in c.items() if is_top}
Y = {n for n, is_top in c.items() if not is_top}
return (X, Y)
[docs]def density(B, nodes):
"""Returns density of bipartite graph B.
Parameters
----------
G : NetworkX graph
nodes: list or container
Nodes in one node set of the bipartite graph.
Returns
-------
d : float
The bipartite density
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.complete_bipartite_graph(3, 2)
>>> X = set([0, 1, 2])
>>> bipartite.density(G, X)
1.0
>>> Y = set([3, 4])
>>> bipartite.density(G, Y)
1.0
Notes
-----
The container of nodes passed as argument must contain all nodes
in one of the two bipartite node sets to avoid ambiguity in the
case of disconnected graphs.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
See Also
--------
color
"""
n = len(B)
m = nx.number_of_edges(B)
nb = len(nodes)
nt = n - nb
if m == 0: # includes cases n==0 and n==1
d = 0.0
else:
if B.is_directed():
d = m / (2.0 * float(nb * nt))
else:
d = m / float(nb * nt)
return d
[docs]def degrees(B, nodes, weight=None):
"""Returns the degrees of the two node sets in the bipartite graph B.
Parameters
----------
G : NetworkX graph
nodes: list or container
Nodes in one node set of the bipartite graph.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight.
If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
Returns
-------
(degX,degY) : tuple of dictionaries
The degrees of the two bipartite sets as dictionaries keyed by node.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.complete_bipartite_graph(3, 2)
>>> Y = set([3, 4])
>>> degX, degY = bipartite.degrees(G, Y)
>>> dict(degX)
{0: 2, 1: 2, 2: 2}
Notes
-----
The container of nodes passed as argument must contain all nodes
in one of the two bipartite node sets to avoid ambiguity in the
case of disconnected graphs.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
See Also
--------
color, density
"""
bottom = set(nodes)
top = set(B) - bottom
return (B.degree(top, weight), B.degree(bottom, weight))