# Copyright (C) 2011 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
# Authors:
# Aric Hagberg <hagberg@lanl.gov>
# Pieter Swart <swart@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Ben Edwards <bedwards@cs.unm.edu>
"""
Graph products.
"""
from itertools import product
import networkx as nx
from networkx.utils import not_implemented_for
__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.nodes[u], H.nodes[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(data=True):
for x, y, d in H.edges(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(data=True):
for x, y, k, d in H.edges(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(data=True, keys=True):
for x, y, d in H.edges(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(data=True, keys=True):
for x, y, k, d in H.edges(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(data=True):
for x, y, d in H.edges(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(data=True):
for x, y, k, d in H.edges(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(data=True, keys=True):
for x, y, d in H.edges(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(data=True, keys=True):
for x, y, k, d in H.edges(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(data=True, keys=True):
for x in H:
yield (u, x), (v, x), k, d
else:
for u, v, d in G.edges(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(data=True, keys=True):
yield (x, u), (x, v), k, d
else:
for x in G:
for u, v, d in H.edges(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(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(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 tensor product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v), (x,y))$ if and only if $(u,x)$ 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()
>>> G.add_node(0, a1=True)
>>> H.add_node('a', a2='Spam')
>>> P = nx.tensor_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_directed_edges_cross_edges(G, H))
if not GH.is_directed():
GH.add_edges_from(_undirected_edges_cross_edges(G, H))
GH.name = "Tensor product(" + G.name + "," + H.name + ")"
return GH
[docs]def cartesian_product(G, H):
r"""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, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v),(x,y))$ if and only if either $u$ is equal to $x$ and
both $v$ and $y$ are adjacent in $H$ or if $v$ is equal to $y$ and both $u$
and $x$ 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()
>>> G.add_node(0, a1=True)
>>> H.add_node('a', a2='Spam')
>>> P = nx.cartesian_product(G, H)
>>> list(P)
[(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.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_edges_cross_nodes(G, H))
GH.add_edges_from(_nodes_cross_edges(G, H))
GH.name = "Cartesian product(" + G.name + "," + H.name + ")"
return GH
[docs]def lexicographic_product(G, H):
r"""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()
>>> G.add_node(0, a1=True)
>>> H.add_node('a', a2='Spam')
>>> P = nx.lexicographic_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
# Edges in G regardless of H designation
GH.add_edges_from(_edges_cross_nodes_and_nodes(G, H))
# For each x in G, only if there is an edge in H
GH.add_edges_from(_nodes_cross_edges(G, H))
GH.name = "Lexicographic product(" + G.name + "," + H.name + ")"
return GH
[docs]def strong_product(G, H):
r"""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()
>>> G.add_node(0, a1=True)
>>> H.add_node('a', a2='Spam')
>>> P = nx.strong_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
"""
GH = _init_product_graph(G, H)
GH.add_nodes_from(_node_product(G, H))
GH.add_edges_from(_nodes_cross_edges(G, H))
GH.add_edges_from(_edges_cross_nodes(G, H))
GH.add_edges_from(_directed_edges_cross_edges(G, H))
if not GH.is_directed():
GH.add_edges_from(_undirected_edges_cross_edges(G, H))
GH.name = "Strong product(" + G.name + "," + H.name + ")"
return GH
[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def power(G, k):
"""Returns the specified power of a graph.
The $k$th power of a simple graph $G$, denoted $G^k$, is a
graph on the same set of nodes in which two distinct nodes $u$ and
$v$ 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 power `k`.
Raises
------
ValueError
If the exponent `k` is not positive.
NetworkXNotImplemented
If `G` is not a simple graph.
Examples
--------
The number of edges will never decrease when taking successive
powers:
>>> G = nx.path_graph(4)
>>> list(nx.power(G, 2).edges)
[(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)]
>>> list(nx.power(G, 3).edges)
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
The `k`th power of a cycle graph on *n* nodes is the complete graph
on *n* nodes, if `k` is at least ``n // 2``:
>>> G = nx.cycle_graph(5)
>>> H = nx.complete_graph(5)
>>> nx.is_isomorphic(nx.power(G, 2), H)
True
>>> G = nx.cycle_graph(8)
>>> H = nx.complete_graph(8)
>>> nx.is_isomorphic(nx.power(G, 4), H)
True
References
----------
.. [1] J. A. Bondy, U. S. R. Murty, *Graph Theory*. Springer, 2008.
Notes
-----
This definition of "power graph" comes from Exercise 3.1.6 of
*Graph Theory* by Bondy and Murty [1]_.
"""
if k <= 0:
raise ValueError('k must be a positive integer')
H = nx.Graph()
H.add_nodes_from(G)
# 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 += 1
H.add_edges_from((n, nbr) for nbr in seen)
return H