469 lines
14 KiB
Python
469 lines
14 KiB
Python
# 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',
|
|
'rooted_product']
|
|
|
|
|
|
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():
|
|
msg = "G and H must be both directed or both undirected"
|
|
raise nx.NetworkXError(msg)
|
|
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
|
|
|
|
|
|
def tensor_product(G, H):
|
|
r"""Returns 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))
|
|
return GH
|
|
|
|
|
|
def cartesian_product(G, H):
|
|
r"""Returns 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
|
|
"""
|
|
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))
|
|
return GH
|
|
|
|
|
|
def lexicographic_product(G, H):
|
|
r"""Returns 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))
|
|
return GH
|
|
|
|
|
|
def strong_product(G, H):
|
|
r"""Returns 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))
|
|
return GH
|
|
|
|
|
|
@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
|
|
|
|
|
|
@not_implemented_for('multigraph')
|
|
def rooted_product(G, H, root):
|
|
""" Return the rooted product of graphs G and H rooted at root in H.
|
|
|
|
A new graph is constructed representing the rooted product of
|
|
the inputted graphs, G and H, with a root in H.
|
|
A rooted product duplicates H for each nodes in G with the root
|
|
of H corresponding to the node in G. Nodes are renamed as the direct
|
|
product of G and H. The result is a subgraph of the cartesian product.
|
|
|
|
Parameters
|
|
----------
|
|
G,H : graph
|
|
A NetworkX graph
|
|
root : node
|
|
A node in H
|
|
|
|
Returns
|
|
-------
|
|
R : The rooted product of G and H with a specified root in H
|
|
|
|
Notes
|
|
-----
|
|
The nodes of R are the Cartesian Product of the nodes of G and H.
|
|
The nodes of G and H are not relabeled.
|
|
"""
|
|
if root not in H:
|
|
raise nx.NetworkXError('root must be a vertex in H')
|
|
|
|
R = nx.Graph()
|
|
R.add_nodes_from(product(G, H))
|
|
|
|
R.add_edges_from(((e[0], root), (e[1], root)) for e in G.edges())
|
|
R.add_edges_from(((g, e[0]), (g, e[1])) for g in G for e in H.edges())
|
|
|
|
return R
|