Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.operators.product

"""
Graph products.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
import networkx as nx
from itertools import product

__author__ = """\n""".join(['Aric Hagberg (hagberg@lanl.gov)',
'Pieter Swart (swart@lanl.gov)',
'Dan Schult(dschult@colgate.edu)'
'Ben Edwards(bedwards@cs.unm.edu)'])

__all__ = ['tensor_product', 'cartesian_product',
'lexicographic_product', 'strong_product', 'power']

def _dict_product(d1, d2):
return dict((k, (d1.get(k), d2.get(k))) for k in set(d1) | set(d2))

# Generators for producting graph products
def _node_product(G, H):
for u, v in product(G, H):
yield ((u, v), _dict_product(G.node[u], H.node[v]))

def _directed_edges_cross_edges(G, H):
if not G.is_multigraph() and not H.is_multigraph():
for u, v, c in G.edges_iter(data=True):
for x, y, d in H.edges_iter(data=True):
yield (u, x), (v, y), _dict_product(c, d)
if not G.is_multigraph() and H.is_multigraph():
for u, v, c in G.edges_iter(data=True):
for x, y, k, d in H.edges_iter(data=True, keys=True):
yield (u, x), (v, y), k, _dict_product(c, d)
if G.is_multigraph() and not H.is_multigraph():
for u, v, k, c in G.edges_iter(data=True, keys=True):
for x, y, d in H.edges_iter(data=True):
yield (u, x), (v, y), k, _dict_product(c, d)
if G.is_multigraph() and H.is_multigraph():
for u, v, j, c in G.edges_iter(data=True, keys=True):
for x, y, k, d in H.edges_iter(data=True, keys=True):
yield (u, x), (v, y), (j, k), _dict_product(c, d)

def _undirected_edges_cross_edges(G, H):
if not G.is_multigraph() and not H.is_multigraph():
for u, v, c in G.edges_iter(data=True):
for x, y, d in H.edges_iter(data=True):
yield (v, x), (u, y), _dict_product(c, d)
if not G.is_multigraph() and H.is_multigraph():
for u, v, c in G.edges_iter(data=True):
for x, y, k, d in H.edges_iter(data=True, keys=True):
yield (v, x), (u, y), k, _dict_product(c, d)
if G.is_multigraph() and not H.is_multigraph():
for u, v, k, c in G.edges_iter(data=True, keys=True):
for x, y, d in H.edges_iter(data=True):
yield (v, x), (u, y), k, _dict_product(c, d)
if G.is_multigraph() and H.is_multigraph():
for u, v, j, c in G.edges_iter(data=True, keys=True):
for x, y, k, d in H.edges_iter(data=True, keys=True):
yield (v, x), (u, y), (j, k), _dict_product(c, d)

def _edges_cross_nodes(G, H):
if G.is_multigraph():
for u, v, k, d in G.edges_iter(data=True, keys=True):
for x in H:
yield (u, x), (v, x), k, d
else:
for u, v, d in G.edges_iter(data=True):
for x in H:
if H.is_multigraph():
yield (u, x), (v, x), None, d
else:
yield (u, x), (v, x), d

def _nodes_cross_edges(G, H):
if H.is_multigraph():
for x in G:
for u, v, k, d in H.edges_iter(data=True, keys=True):
yield (x, u), (x, v), k, d
else:
for x in G:
for u, v, d in H.edges_iter(data=True):
if G.is_multigraph():
yield (x, u), (x, v), None, d
else:
yield (x, u), (x, v), d

def _edges_cross_nodes_and_nodes(G, H):
if G.is_multigraph():
for u, v, k, d in G.edges_iter(data=True, keys=True):
for x in H:
for y in H:
yield (u, x), (v, y), k, d
else:
for u, v, d in G.edges_iter(data=True):
for x in H:
for y in H:
if H.is_multigraph():
yield (u, x), (v, y), None, d
else:
yield (u, x), (v, y), d

def _init_product_graph(G, H):
if not G.is_directed() == H.is_directed():
raise nx.NetworkXError("G and H must be both directed or",
"both undirected")
if G.is_multigraph() or H.is_multigraph():
GH = nx.MultiGraph()
else:
GH = nx.Graph()
if G.is_directed():
GH = GH.to_directed()
return GH

[docs]def tensor_product(G, H):
r"""Return the tensor product of G and H.

The tensor product P of the graphs G and H has a node set that
is the Cartesian product of the node sets, :math:V(P)=V(G) \times V(H).
P has an edge ((u,v),(x,y)) if and only if (u,x) is an edge in G
and (v,y) is an edge in H.

Tensor product is sometimes also referred to as the categorical product,
direct product, cardinal product or conjunction.

Parameters
----------
G, H: graphs
Networkx graphs.

Returns
-------
P: NetworkX graph
The tensor product of G and H. P will be a multi-graph if either G
or H is a multi-graph, will be a directed if G and H are directed,
and undirected if G and H are undirected.

Raises
------
NetworkXError
If G and H are not both directed or both undirected.

Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.

Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> P = nx.tensor_product(G,H)
>>> P.nodes()
[(0, 'a')]

Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
if not GH.is_directed():
GH.name = "Tensor product(" + G.name + "," + H.name + ")"
return GH

[docs]def cartesian_product(G, H):
"""Return the Cartesian product of G and H.

The Cartesian product P of the graphs G and H has a node set that
is the Cartesian product of the node sets, :math:V(P)=V(G) \times V(H).
P has an edge ((u,v),(x,y)) if and only if either u is equal to x and
v & y are adjacent in H or if v is equal to y and u & x are adjacent in G.

Parameters
----------
G, H: graphs
Networkx graphs.

Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.

Raises
------
NetworkXError
If G and H are not both directed or both undirected.

Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.

Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> P = nx.cartesian_product(G,H)
>>> P.nodes()
[(0, 'a')]

Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
if not G.is_directed() == H.is_directed():
raise nx.NetworkXError("G and H must be both directed or",
"both undirected")
GH = _init_product_graph(G, H)
GH.name = "Cartesian product(" + G.name + "," + H.name + ")"
return GH

[docs]def lexicographic_product(G, H):
"""Return the lexicographic product of G and H.

The lexicographical product P of the graphs G and H has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G
or u==v and (x,y) is an edge in H.

Parameters
----------
G, H: graphs
Networkx graphs.

Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.

Raises
------
NetworkXError
If G and H are not both directed or both undirected.

Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.

Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> P = nx.lexicographic_product(G,H)
>>> P.nodes()
[(0, 'a')]

Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
# Edges in G regardless of H designation
# For each x in G, only if there is an edge in H
GH.name = "Lexicographic product(" + G.name + "," + H.name + ")"
return GH

[docs]def strong_product(G, H):
"""Return the strong product of G and H.

The strong product P of the graphs G and H has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
P has an edge ((u,v),(x,y)) if and only if
u==v and (x,y) is an edge in H, or
x==y and (u,v) is an edge in G, or
(u,v) is an edge in G and (x,y) is an edge in H.

Parameters
----------
G, H: graphs
Networkx graphs.

Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.

Raises
------
NetworkXError
If G and H are not both directed or both undirected.

Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.

Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> P = nx.strong_product(G,H)
>>> P.nodes()
[(0, 'a')]

Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
if not GH.is_directed():
GH.name = "Strong product(" + G.name + "," + H.name + ")"
return GH

[docs]def power(G, k):
"""Returns the specified power of a graph.

The k-th power of a simple graph G = (V, E) is the graph
G^k whose vertex set is V,  two distinct vertices u,v are
adjacent in G^k if and only if the shortest path
distance between u and v in G is at most k.

Parameters
----------
G: graph
A NetworkX simple graph object.
k: positive integer
The power to which to raise the graph G.

Returns
-------
NetworkX simple graph
G to the k-th power.

Raises
------
:exc:ValueError
If the exponent k is not positive.

NetworkXError
If G is not a simple graph.

Examples
--------
>>> G = nx.path_graph(4)
>>> nx.power(G,2).edges()
[(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)]
>>> nx.power(G,3).edges()
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]

A complete graph of order n is returned if *k* is greater than equal to n/2
for a cycle graph of even order n, and if *k* is greater than equal to
(n-1)/2 for a cycle graph of odd order.

>>> G = nx.cycle_graph(5)
>>> nx.power(G,2).edges() == nx.complete_graph(5).edges()
True
>>> G = nx.cycle_graph(8)
>>> nx.power(G,4).edges() == nx.complete_graph(8).edges()
True

References
----------
.. [1] J. A. Bondy, U. S. R. Murty, Graph Theory. Springer, 2008.

Notes
-----
Exercise 3.1.6 of *Graph Theory* by J. A. Bondy and U. S. R. Murty [1]_.

"""
if k <= 0:
raise ValueError('k must be a positive integer')
if G.is_multigraph() or G.is_directed():
raise nx.NetworkXError("G should be a simple graph")
H = nx.Graph()
# update BFS code to ignore self loops.
for n in G:
seen = {}                  # level (number of hops) when seen in BFS
level = 1                  # the current level
nextlevel = G[n]
while nextlevel:
thislevel = nextlevel  # advance to next level
nextlevel = {}         # and start a new list (fringe)
for v in thislevel:
if v == n:         # avoid self loop
continue
if v not in seen:
seen[v] = level         # set the level of vertex v
nextlevel.update(G[v])  # add neighbors of v
if (k <= level):
break
level = level + 1
H.add_edges_from((n, nbr) for nbr in seen)
return H