# Copyright (C) 2004-2018 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)
"""Generators for some classic graphs.
The typical graph generator is called as follows:
>>> G = nx.complete_graph(100)
returning the complete graph on n nodes labeled 0, .., 99
as a simple graph. Except for empty_graph, all the generators
in this module return a Graph class (i.e. a simple, undirected graph).
"""
from __future__ import division
import itertools
import networkx as nx
from networkx.classes import Graph
from networkx.exception import NetworkXError
from networkx.utils import accumulate
from networkx.utils import nodes_or_number
from networkx.utils import pairwise
__all__ = ['balanced_tree',
'barbell_graph',
'complete_graph',
'complete_multipartite_graph',
'circular_ladder_graph',
'circulant_graph',
'cycle_graph',
'dorogovtsev_goltsev_mendes_graph',
'empty_graph',
'full_rary_tree',
'ladder_graph',
'lollipop_graph',
'null_graph',
'path_graph',
'star_graph',
'trivial_graph',
'turan_graph',
'wheel_graph']
# -------------------------------------------------------------------
# Some Classic Graphs
# -------------------------------------------------------------------
def _tree_edges(n, r):
# helper function for trees
# yields edges in rooted tree at 0 with n nodes and branching ratio r
nodes = iter(range(n))
parents = [next(nodes)] # stack of max length r
while parents:
source = parents.pop(0)
for i in range(r):
try:
target = next(nodes)
parents.append(target)
yield source, target
except StopIteration:
break
def full_rary_tree(r, n, create_using=None):
"""Creates a full r-ary tree of n vertices.
Sometimes called a k-ary, n-ary, or m-ary tree.
"... all non-leaf vertices have exactly r children and all levels
are full except for some rightmost position of the bottom level
(if a leaf at the bottom level is missing, then so are all of the
leaves to its right." [1]_
Parameters
----------
r : int
branching factor of the tree
n : int
Number of nodes in the tree
create_using : Graph, optional (default None)
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Returns
-------
G : networkx Graph
An r-ary tree with n nodes
References
----------
.. [1] An introduction to data structures and algorithms,
James Andrew Storer, Birkhauser Boston 2001, (page 225).
"""
G = empty_graph(n, create_using)
G.add_edges_from(_tree_edges(n, r))
return G
[docs]def balanced_tree(r, h, create_using=None):
"""Return the perfectly balanced `r`-ary tree of height `h`.
Parameters
----------
r : int
Branching factor of the tree; each node will have `r`
children.
h : int
Height of the tree.
create_using : Graph, optional (default None)
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Returns
-------
G : NetworkX graph
A balanced `r`-ary tree of height `h`.
Notes
-----
This is the rooted tree where all leaves are at distance `h` from
the root. The root has degree `r` and all other internal nodes
have degree `r + 1`.
Node labels are integers, starting from zero.
A balanced tree is also known as a *complete r-ary tree*.
"""
# The number of nodes in the balanced tree is `1 + r + ... + r^h`,
# which is computed by using the closed-form formula for a geometric
# sum with ratio `r`. In the special case that `r` is 1, the number
# of nodes is simply `h + 1` (since the tree is actually a path
# graph).
if r == 1:
n = h + 1
else:
# This must be an integer if both `r` and `h` are integers. If
# they are not, we force integer division anyway.
n = (1 - r ** (h + 1)) // (1 - r)
return full_rary_tree(r, n, create_using=create_using)
[docs]def barbell_graph(m1, m2, create_using=None):
"""Return the Barbell Graph: two complete graphs connected by a path.
For $m1 > 1$ and $m2 >= 0$.
Two identical complete graphs $K_{m1}$ form the left and right bells,
and are connected by a path $P_{m2}$.
The `2*m1+m2` nodes are numbered
`0, ..., m1-1` for the left barbell,
`m1, ..., m1+m2-1` for the path,
and `m1+m2, ..., 2*m1+m2-1` for the right barbell.
The 3 subgraphs are joined via the edges `(m1-1, m1)` and
`(m1+m2-1, m1+m2)`. If `m2=0`, this is merely two complete
graphs joined together.
This graph is an extremal example in David Aldous
and Jim Fill's e-text on Random Walks on Graphs.
"""
if create_using is not None and create_using.is_directed():
raise NetworkXError("Directed Graph not supported")
if m1 < 2:
raise NetworkXError(
"Invalid graph description, m1 should be >=2")
if m2 < 0:
raise NetworkXError(
"Invalid graph description, m2 should be >=0")
# left barbell
G = complete_graph(m1, create_using)
# connecting path
G.add_nodes_from(range(m1, m1 + m2 - 1))
if m2 > 1:
G.add_edges_from(pairwise(range(m1, m1 + m2)))
# right barbell
G.add_edges_from((u, v) for u in range(m1 + m2, 2 * m1 + m2)
for v in range(u + 1, 2 * m1 + m2))
# connect it up
G.add_edge(m1 - 1, m1)
if m2 > 0:
G.add_edge(m1 + m2 - 1, m1 + m2)
return G
[docs]@nodes_or_number(0)
def complete_graph(n, create_using=None):
""" Return the complete graph `K_n` with n nodes.
Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from range(n).
If n is a container of nodes, those nodes appear in the graph.
create_using : Graph, optional (default None)
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Examples
--------
>>> G = nx.complete_graph(9)
>>> len(G)
9
>>> G.size()
36
>>> G = nx.complete_graph(range(11, 14))
>>> list(G.nodes())
[11, 12, 13]
>>> G = nx.complete_graph(4, nx.DiGraph())
>>> G.is_directed()
True
"""
n_name, nodes = n
G = empty_graph(n_name, create_using)
if len(nodes) > 1:
if G.is_directed():
edges = itertools.permutations(nodes, 2)
else:
edges = itertools.combinations(nodes, 2)
G.add_edges_from(edges)
return G
[docs]def circular_ladder_graph(n, create_using=None):
"""Return the circular ladder graph $CL_n$ of length n.
$CL_n$ consists of two concentric n-cycles in which
each of the n pairs of concentric nodes are joined by an edge.
Node labels are the integers 0 to n-1
"""
G = ladder_graph(n, create_using)
G.add_edge(0, n - 1)
G.add_edge(n, 2 * n - 1)
return G
def circulant_graph(n, offsets, create_using=None):
"""Generates the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ vertices.
Returns
-------
The graph $Ci_n(x_1, ..., x_m)$ consisting of $n$ vertices $0, ..., n-1$ such
that the vertex with label $i$ is connected to the vertices labelled $(i + x)$
and $(i - x)$, for all $x$ in $x_1$ up to $x_m$, with the indices taken modulo $n$.
Parameters
----------
n : integer
The number of vertices the generated graph is to contain.
offsets : list of integers
A list of vertex offsets, $x_1$ up to $x_m$, as described above.
create_using : Graph, optional (default None)
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Examples
--------
Many well-known graph families are subfamilies of the circulant graphs;
for example, to generate the cycle graph on n points, we connect every
vertex to every other at offset plus or minus one. For n = 10,
>>> import networkx
>>> G = networkx.generators.classic.circulant_graph(10, [1])
>>> edges = [
... (0, 9), (0, 1), (1, 2), (2, 3), (3, 4),
... (4, 5), (5, 6), (6, 7), (7, 8), (8, 9)]
...
>>> sorted(edges) == sorted(G.edges())
True
Similarly, we can generate the complete graph on 5 points with the set of
offsets [1, 2]:
>>> G = networkx.generators.classic.circulant_graph(5, [1, 2])
>>> edges = [
... (0, 1), (0, 2), (0, 3), (0, 4), (1, 2),
... (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]
...
>>> sorted(edges) == sorted(G.edges())
True
"""
G = empty_graph(n, create_using)
for i in range(n):
for j in offsets:
G.add_edge(i, (i - j) % n)
G.add_edge(i, (i + j) % n)
return G
[docs]@nodes_or_number(0)
def cycle_graph(n, create_using=None):
"""Return the cycle graph $C_n$ of cyclically connected nodes.
$C_n$ is a path with its two end-nodes connected.
Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Notes
-----
If create_using is directed, the direction is in increasing order.
"""
n_orig, nodes = n
G = empty_graph(nodes, create_using)
G.add_edges_from(pairwise(nodes))
G.add_edge(nodes[-1], nodes[0])
return G
[docs]def dorogovtsev_goltsev_mendes_graph(n, create_using=None):
"""Return the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph.
n is the generation.
See: arXiv:/cond-mat/0112143 by Dorogovtsev, Goltsev and Mendes.
"""
if create_using is not None:
if create_using.is_directed():
raise NetworkXError("Directed Graph not supported")
if create_using.is_multigraph():
raise NetworkXError("Multigraph not supported")
G = empty_graph(0, create_using)
G.add_edge(0, 1)
if n == 0:
return G
new_node = 2 # next node to be added
for i in range(1, n + 1): # iterate over number of generations.
last_generation_edges = list(G.edges())
number_of_edges_in_last_generation = len(last_generation_edges)
for j in range(0, number_of_edges_in_last_generation):
G.add_edge(new_node, last_generation_edges[j][0])
G.add_edge(new_node, last_generation_edges[j][1])
new_node += 1
return G
[docs]@nodes_or_number(0)
def empty_graph(n=0, create_using=None):
"""Return the empty graph with n nodes and zero edges.
Parameters
----------
n : int or iterable container of nodes (default = 0)
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Examples
--------
>>> G = nx.empty_graph(10)
>>> G.number_of_nodes()
10
>>> G.number_of_edges()
0
>>> G = nx.empty_graph("ABC")
>>> G.number_of_nodes()
3
>>> sorted(G)
['A', 'B', 'C']
Notes
-----
The variable create_using should point to a "graph"-like object that
will be cleared (nodes and edges will be removed) and refitted as
an empty "graph" with nodes specified in n. This capability
is useful for specifying the class-nature of the resulting empty
"graph" (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).
The variable create_using has two main uses:
Firstly, the variable create_using can be used to create an
empty digraph, multigraph, etc. For example,
>>> n = 10
>>> G = nx.empty_graph(n, create_using=nx.DiGraph())
will create an empty digraph on n nodes.
Secondly, one can pass an existing graph (digraph, multigraph,
etc.) via create_using. For example, if G is an existing graph
(resp. digraph, multigraph, etc.), then empty_graph(n, create_using=G)
will empty G (i.e. delete all nodes and edges using G.clear())
and then add n nodes and zero edges, and return the modified graph.
See also create_empty_copy(G).
"""
if create_using is None:
# default empty graph is a simple graph
G = Graph()
else:
G = create_using
G.clear()
n_name, nodes = n
G.add_nodes_from(nodes)
return G
[docs]def ladder_graph(n, create_using=None):
"""Return the Ladder graph of length n.
This is two paths of n nodes, with
each pair connected by a single edge.
Node labels are the integers 0 to 2*n - 1.
"""
if create_using is not None and create_using.is_directed():
raise NetworkXError("Directed Graph not supported")
G = empty_graph(2 * n, create_using)
G.add_edges_from(pairwise(range(n)))
G.add_edges_from(pairwise(range(n, 2 * n)))
G.add_edges_from((v, v + n) for v in range(n))
return G
[docs]@nodes_or_number([0, 1])
def lollipop_graph(m, n, create_using=None):
"""Return the Lollipop Graph; `K_m` connected to `P_n`.
This is the Barbell Graph without the right barbell.
Parameters
----------
m, n : int or iterable container of nodes (default = 0)
If an integer, nodes are from `range(m)` and `range(m,m+n)`.
If a container, the entries are the coordinate of the node.
The nodes for m appear in the complete graph $K_m$ and the nodes
for n appear in the path $P_n$
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Notes
-----
The 2 subgraphs are joined via an edge (m-1, m).
If n=0, this is merely a complete graph.
(This graph is an extremal example in David Aldous and Jim
Fill's etext on Random Walks on Graphs.)
"""
m, m_nodes = m
n, n_nodes = n
M = len(m_nodes)
N = len(n_nodes)
if isinstance(m, int):
n_nodes = [len(m_nodes) + i for i in n_nodes]
if create_using is not None and create_using.is_directed():
raise NetworkXError("Directed Graph not supported")
if M < 2:
raise NetworkXError(
"Invalid graph description, m should be >=2")
if N < 0:
raise NetworkXError(
"Invalid graph description, n should be >=0")
# the ball
G = complete_graph(m_nodes, create_using)
# the stick
G.add_nodes_from(n_nodes)
if N > 1:
G.add_edges_from(pairwise(n_nodes))
# connect ball to stick
if M > 0 and N > 0:
G.add_edge(m_nodes[-1], n_nodes[0])
return G
[docs]def null_graph(create_using=None):
"""Return the Null graph with no nodes or edges.
See empty_graph for the use of create_using.
"""
G = empty_graph(0, create_using)
return G
[docs]@nodes_or_number(0)
def path_graph(n, create_using=None):
"""Return the Path graph `P_n` of linearly connected nodes.
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
"""
n_name, nodes = n
G = empty_graph(nodes, create_using)
G.add_edges_from(pairwise(nodes))
return G
[docs]@nodes_or_number(0)
def star_graph(n, create_using=None):
""" Return the star graph
The star graph consists of one center node connected to n outer nodes.
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n_name, nodes = n
if isinstance(n_name, int):
nodes = nodes + [n_name] # there should be n+1 nodes
first = nodes[0]
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
G.add_edges_from((first, v) for v in nodes[1:])
return G
[docs]def trivial_graph(create_using=None):
""" Return the Trivial graph with one node (with label 0) and no edges.
"""
G = empty_graph(1, create_using)
return G
[docs]def turan_graph(n, r):
r""" Return the Turan Graph
The Turan Graph is a complete multipartite graph on $n$ vertices
with $r$ disjoint subsets. It is the graph with the edges for any graph with
$n$ vertices and $r$ disjoint subsets.
Given $n$ and $r$, we generate a complete multipartite graph with
$r-(n \mod r)$ partitions of size $n/r$, rounded down, and
$n \mod r$ partitions of size $n/r+1$, rounded down.
Parameters
----------
n : int
The number of vertices.
r : int
The number of partitions.
Must be less than or equal to n.
Notes
-----
Must satisfy $1 <= r <= n$.
The graph has $(r-1)(n^2)/(2r)$ edges, rounded down.
"""
if not 1 <= r <= n:
raise NetworkXError("Must satisfy 1 <= r <= n")
partitions = [n // r] * (r - (n % r)) + [n // r + 1] * (n % r)
G = complete_multipartite_graph(*partitions)
return G
[docs]@nodes_or_number(0)
def wheel_graph(n, create_using=None):
""" Return the wheel graph
The wheel graph consists of a hub node connected to a cycle of (n-1) nodes.
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
create_using : Graph, optional (default Graph())
If provided this graph is cleared of nodes and edges and filled
with the new graph. Usually used to set the type of the graph.
Node labels are the integers 0 to n - 1.
"""
n_name, nodes = n
if n_name == 0:
G = empty_graph(0, create_using=create_using)
return G
G = star_graph(nodes, create_using)
if len(G) > 2:
G.add_edges_from(pairwise(nodes[1:]))
G.add_edge(nodes[-1], nodes[1])
return G
[docs]def complete_multipartite_graph(*subset_sizes):
"""Returns the complete multipartite graph with the specified subset sizes.
Parameters
----------
subset_sizes : tuple of integers or tuple of node iterables
The arguments can either all be integer number of nodes or they
can all be iterables of nodes. If integers, they represent the
number of vertices in each subset of the multipartite graph.
If iterables, each is used to create the nodes for that subset.
The length of subset_sizes is the number of subsets.
Returns
-------
G : NetworkX Graph
Returns the complete multipartite graph with the specified subsets.
For each node, the node attribute 'subset' is an integer
indicating which subset contains the node.
Examples
--------
Creating a complete tripartite graph, with subsets of one, two, and three
vertices, respectively.
>>> import networkx as nx
>>> G = nx.complete_multipartite_graph(1, 2, 3)
>>> [G.nodes[u]['subset'] for u in G]
[0, 1, 1, 2, 2, 2]
>>> list(G.edges(0))
[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)]
>>> list(G.edges(2))
[(2, 0), (2, 3), (2, 4), (2, 5)]
>>> list(G.edges(4))
[(4, 0), (4, 1), (4, 2)]
>>> G = nx.complete_multipartite_graph('a', 'bc', 'def')
>>> [G.nodes[u]['subset'] for u in sorted(G)]
[0, 1, 1, 2, 2, 2]
Notes
-----
This function generalizes several other graph generator functions.
- If no subset sizes are given, this returns the null graph.
- If a single subset size `n` is given, this returns the empty graph on
`n` nodes.
- If two subset sizes `m` and `n` are given, this returns the complete
bipartite graph on `m + n` nodes.
- If subset sizes `1` and `n` are given, this returns the star graph on
`n + 1` nodes.
See also
--------
complete_bipartite_graph
"""
# The complete multipartite graph is an undirected simple graph.
G = Graph()
if len(subset_sizes) == 0:
return G
# set up subsets of nodes
try:
extents = pairwise(accumulate((0,) + subset_sizes))
subsets = [range(start, end) for start, end in extents]
except TypeError:
subsets = subset_sizes
# add nodes with subset attribute
# while checking that ints are not mixed with iterables
try:
for (i, subset) in enumerate(subsets):
G.add_nodes_from(subset, subset=i)
except TypeError:
raise NetworkXError("Arguments must be all ints or all iterables")
# Across subsets, all vertices should be adjacent.
# We can use itertools.combinations() because undirected.
for subset1, subset2 in itertools.combinations(subsets, 2):
G.add_edges_from(itertools.product(subset1, subset2))
return G