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766 lines
23 KiB
766 lines
23 KiB
"""Generators for some classic graphs.
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The typical graph generator is called as follows:
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>>> G = nx.complete_graph(100)
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returning the complete graph on n nodes labeled 0, .., 99
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as a simple graph. Except for empty_graph, all the generators
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in this module return a Graph class (i.e. a simple, undirected graph).
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"""
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import itertools
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import networkx as nx
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from networkx.classes import Graph
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from networkx.exception import NetworkXError
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from itertools import accumulate
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from networkx.utils import nodes_or_number
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from networkx.utils import pairwise
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__all__ = [
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"balanced_tree",
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"barbell_graph",
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"binomial_tree",
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"complete_graph",
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"complete_multipartite_graph",
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"circular_ladder_graph",
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"circulant_graph",
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"cycle_graph",
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"dorogovtsev_goltsev_mendes_graph",
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"empty_graph",
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"full_rary_tree",
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"ladder_graph",
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"lollipop_graph",
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"null_graph",
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"path_graph",
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"star_graph",
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"trivial_graph",
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"turan_graph",
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"wheel_graph",
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]
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# -------------------------------------------------------------------
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# Some Classic Graphs
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# -------------------------------------------------------------------
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def _tree_edges(n, r):
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if n == 0:
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return
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# helper function for trees
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# yields edges in rooted tree at 0 with n nodes and branching ratio r
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nodes = iter(range(n))
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parents = [next(nodes)] # stack of max length r
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while parents:
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source = parents.pop(0)
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for i in range(r):
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try:
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target = next(nodes)
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parents.append(target)
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yield source, target
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except StopIteration:
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break
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def full_rary_tree(r, n, create_using=None):
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"""Creates a full r-ary tree of n vertices.
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Sometimes called a k-ary, n-ary, or m-ary tree.
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"... all non-leaf vertices have exactly r children and all levels
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are full except for some rightmost position of the bottom level
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(if a leaf at the bottom level is missing, then so are all of the
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leaves to its right." [1]_
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Parameters
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----------
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r : int
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branching factor of the tree
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n : int
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Number of nodes in the tree
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Returns
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-------
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G : networkx Graph
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An r-ary tree with n nodes
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References
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----------
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.. [1] An introduction to data structures and algorithms,
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James Andrew Storer, Birkhauser Boston 2001, (page 225).
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"""
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G = empty_graph(n, create_using)
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G.add_edges_from(_tree_edges(n, r))
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return G
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def balanced_tree(r, h, create_using=None):
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"""Returns the perfectly balanced `r`-ary tree of height `h`.
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Parameters
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----------
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r : int
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Branching factor of the tree; each node will have `r`
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children.
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h : int
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Height of the tree.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Returns
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-------
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G : NetworkX graph
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A balanced `r`-ary tree of height `h`.
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Notes
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-----
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This is the rooted tree where all leaves are at distance `h` from
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the root. The root has degree `r` and all other internal nodes
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have degree `r + 1`.
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Node labels are integers, starting from zero.
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A balanced tree is also known as a *complete r-ary tree*.
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"""
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# The number of nodes in the balanced tree is `1 + r + ... + r^h`,
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# which is computed by using the closed-form formula for a geometric
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# sum with ratio `r`. In the special case that `r` is 1, the number
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# of nodes is simply `h + 1` (since the tree is actually a path
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# graph).
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if r == 1:
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n = h + 1
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else:
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# This must be an integer if both `r` and `h` are integers. If
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# they are not, we force integer division anyway.
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n = (1 - r ** (h + 1)) // (1 - r)
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return full_rary_tree(r, n, create_using=create_using)
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def barbell_graph(m1, m2, create_using=None):
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"""Returns the Barbell Graph: two complete graphs connected by a path.
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For $m1 > 1$ and $m2 >= 0$.
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Two identical complete graphs $K_{m1}$ form the left and right bells,
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and are connected by a path $P_{m2}$.
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The `2*m1+m2` nodes are numbered
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`0, ..., m1-1` for the left barbell,
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`m1, ..., m1+m2-1` for the path,
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and `m1+m2, ..., 2*m1+m2-1` for the right barbell.
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The 3 subgraphs are joined via the edges `(m1-1, m1)` and
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`(m1+m2-1, m1+m2)`. If `m2=0`, this is merely two complete
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graphs joined together.
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This graph is an extremal example in David Aldous
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and Jim Fill's e-text on Random Walks on Graphs.
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"""
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if m1 < 2:
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raise NetworkXError("Invalid graph description, m1 should be >=2")
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if m2 < 0:
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raise NetworkXError("Invalid graph description, m2 should be >=0")
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# left barbell
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G = complete_graph(m1, create_using)
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if G.is_directed():
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raise NetworkXError("Directed Graph not supported")
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# connecting path
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G.add_nodes_from(range(m1, m1 + m2 - 1))
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if m2 > 1:
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G.add_edges_from(pairwise(range(m1, m1 + m2)))
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# right barbell
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G.add_edges_from(
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(u, v) for u in range(m1 + m2, 2 * m1 + m2) for v in range(u + 1, 2 * m1 + m2)
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)
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# connect it up
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G.add_edge(m1 - 1, m1)
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if m2 > 0:
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G.add_edge(m1 + m2 - 1, m1 + m2)
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return G
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def binomial_tree(n):
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"""Returns the Binomial Tree of order n.
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The binomial tree of order 0 consists of a single vertex. A binomial tree of order k
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is defined recursively by linking two binomial trees of order k-1: the root of one is
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the leftmost child of the root of the other.
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Parameters
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----------
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n : int
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Order of the binomial tree.
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Returns
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-------
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G : NetworkX graph
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A binomial tree of $2^n$ vertices and $2^n - 1$ edges.
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"""
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G = nx.empty_graph(1)
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N = 1
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for i in range(n):
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edges = [(u + N, v + N) for (u, v) in G.edges]
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G.add_edges_from(edges)
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G.add_edge(0, N)
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N *= 2
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return G
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@nodes_or_number(0)
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def complete_graph(n, create_using=None):
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""" Return the complete graph `K_n` with n nodes.
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Parameters
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----------
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n : int or iterable container of nodes
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If n is an integer, nodes are from range(n).
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If n is a container of nodes, those nodes appear in the graph.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Examples
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--------
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>>> G = nx.complete_graph(9)
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>>> len(G)
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9
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>>> G.size()
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36
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>>> G = nx.complete_graph(range(11, 14))
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>>> list(G.nodes())
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[11, 12, 13]
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>>> G = nx.complete_graph(4, nx.DiGraph())
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>>> G.is_directed()
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True
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"""
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n_name, nodes = n
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G = empty_graph(n_name, create_using)
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if len(nodes) > 1:
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if G.is_directed():
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edges = itertools.permutations(nodes, 2)
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else:
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edges = itertools.combinations(nodes, 2)
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G.add_edges_from(edges)
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return G
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def circular_ladder_graph(n, create_using=None):
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"""Returns the circular ladder graph $CL_n$ of length n.
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$CL_n$ consists of two concentric n-cycles in which
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each of the n pairs of concentric nodes are joined by an edge.
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Node labels are the integers 0 to n-1
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"""
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G = ladder_graph(n, create_using)
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G.add_edge(0, n - 1)
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G.add_edge(n, 2 * n - 1)
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return G
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def circulant_graph(n, offsets, create_using=None):
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"""Generates the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ vertices.
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Returns
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-------
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The graph $Ci_n(x_1, ..., x_m)$ consisting of $n$ vertices $0, ..., n-1$ such
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that the vertex with label $i$ is connected to the vertices labelled $(i + x)$
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and $(i - x)$, for all $x$ in $x_1$ up to $x_m$, with the indices taken modulo $n$.
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Parameters
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----------
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n : integer
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The number of vertices the generated graph is to contain.
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offsets : list of integers
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A list of vertex offsets, $x_1$ up to $x_m$, as described above.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Examples
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--------
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Many well-known graph families are subfamilies of the circulant graphs;
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for example, to generate the cycle graph on n points, we connect every
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vertex to every other at offset plus or minus one. For n = 10,
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>>> import networkx
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>>> G = networkx.generators.classic.circulant_graph(10, [1])
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>>> edges = [
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... (0, 9),
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... (0, 1),
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... (1, 2),
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... (2, 3),
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... (3, 4),
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... (4, 5),
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... (5, 6),
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... (6, 7),
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... (7, 8),
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... (8, 9),
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... ]
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...
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>>> sorted(edges) == sorted(G.edges())
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True
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Similarly, we can generate the complete graph on 5 points with the set of
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offsets [1, 2]:
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>>> G = networkx.generators.classic.circulant_graph(5, [1, 2])
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>>> edges = [
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... (0, 1),
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... (0, 2),
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... (0, 3),
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... (0, 4),
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... (1, 2),
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... (1, 3),
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... (1, 4),
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... (2, 3),
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... (2, 4),
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... (3, 4),
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... ]
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...
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>>> sorted(edges) == sorted(G.edges())
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True
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"""
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G = empty_graph(n, create_using)
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for i in range(n):
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for j in offsets:
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G.add_edge(i, (i - j) % n)
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G.add_edge(i, (i + j) % n)
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return G
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@nodes_or_number(0)
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def cycle_graph(n, create_using=None):
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"""Returns the cycle graph $C_n$ of cyclically connected nodes.
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$C_n$ is a path with its two end-nodes connected.
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Parameters
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----------
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n : int or iterable container of nodes
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If n is an integer, nodes are from `range(n)`.
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If n is a container of nodes, those nodes appear in the graph.
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Notes
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-----
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If create_using is directed, the direction is in increasing order.
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"""
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n_orig, nodes = n
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G = empty_graph(nodes, create_using)
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G.add_edges_from(pairwise(nodes))
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G.add_edge(nodes[-1], nodes[0])
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return G
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def dorogovtsev_goltsev_mendes_graph(n, create_using=None):
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"""Returns the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph.
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n is the generation.
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See: arXiv:/cond-mat/0112143 by Dorogovtsev, Goltsev and Mendes.
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"""
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G = empty_graph(0, create_using)
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if G.is_directed():
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raise NetworkXError("Directed Graph not supported")
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if G.is_multigraph():
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raise NetworkXError("Multigraph not supported")
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G.add_edge(0, 1)
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if n == 0:
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return G
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new_node = 2 # next node to be added
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for i in range(1, n + 1): # iterate over number of generations.
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last_generation_edges = list(G.edges())
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number_of_edges_in_last_generation = len(last_generation_edges)
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for j in range(0, number_of_edges_in_last_generation):
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G.add_edge(new_node, last_generation_edges[j][0])
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G.add_edge(new_node, last_generation_edges[j][1])
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new_node += 1
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return G
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@nodes_or_number(0)
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def empty_graph(n=0, create_using=None, default=nx.Graph):
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"""Returns the empty graph with n nodes and zero edges.
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Parameters
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----------
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n : int or iterable container of nodes (default = 0)
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If n is an integer, nodes are from `range(n)`.
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If n is a container of nodes, those nodes appear in the graph.
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create_using : Graph Instance, Constructor or None
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Indicator of type of graph to return.
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If a Graph-type instance, then clear and use it.
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If None, use the `default` constructor.
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If a constructor, call it to create an empty graph.
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default : Graph constructor (optional, default = nx.Graph)
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The constructor to use if create_using is None.
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If None, then nx.Graph is used.
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This is used when passing an unknown `create_using` value
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through your home-grown function to `empty_graph` and
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you want a default constructor other than nx.Graph.
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Examples
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--------
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>>> G = nx.empty_graph(10)
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>>> G.number_of_nodes()
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10
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>>> G.number_of_edges()
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0
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>>> G = nx.empty_graph("ABC")
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>>> G.number_of_nodes()
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3
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>>> sorted(G)
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['A', 'B', 'C']
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Notes
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-----
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The variable create_using should be a Graph Constructor or a
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"graph"-like object. Constructors, e.g. `nx.Graph` or `nx.MultiGraph`
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will be used to create the returned graph. "graph"-like objects
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will be cleared (nodes and edges will be removed) and refitted as
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an empty "graph" with nodes specified in n. This capability
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is useful for specifying the class-nature of the resulting empty
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"graph" (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).
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The variable create_using has three main uses:
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Firstly, the variable create_using can be used to create an
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empty digraph, multigraph, etc. For example,
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>>> n = 10
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>>> G = nx.empty_graph(n, create_using=nx.DiGraph)
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will create an empty digraph on n nodes.
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Secondly, one can pass an existing graph (digraph, multigraph,
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etc.) via create_using. For example, if G is an existing graph
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(resp. digraph, multigraph, etc.), then empty_graph(n, create_using=G)
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will empty G (i.e. delete all nodes and edges using G.clear())
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and then add n nodes and zero edges, and return the modified graph.
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Thirdly, when constructing your home-grown graph creation function
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you can use empty_graph to construct the graph by passing a user
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defined create_using to empty_graph. In this case, if you want the
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default constructor to be other than nx.Graph, specify `default`.
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>>> def mygraph(n, create_using=None):
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... G = nx.empty_graph(n, create_using, nx.MultiGraph)
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... G.add_edges_from([(0, 1), (0, 1)])
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... return G
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>>> G = mygraph(3)
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>>> G.is_multigraph()
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True
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>>> G = mygraph(3, nx.Graph)
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>>> G.is_multigraph()
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False
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See also create_empty_copy(G).
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"""
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if create_using is None:
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G = default()
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elif hasattr(create_using, "_adj"):
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# create_using is a NetworkX style Graph
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create_using.clear()
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G = create_using
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else:
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# try create_using as constructor
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G = create_using()
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n_name, nodes = n
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G.add_nodes_from(nodes)
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return G
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def ladder_graph(n, create_using=None):
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"""Returns the Ladder graph of length n.
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This is two paths of n nodes, with
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each pair connected by a single edge.
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Node labels are the integers 0 to 2*n - 1.
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"""
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G = empty_graph(2 * n, create_using)
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if G.is_directed():
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raise NetworkXError("Directed Graph not supported")
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G.add_edges_from(pairwise(range(n)))
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G.add_edges_from(pairwise(range(n, 2 * n)))
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G.add_edges_from((v, v + n) for v in range(n))
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return G
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@nodes_or_number([0, 1])
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def lollipop_graph(m, n, create_using=None):
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"""Returns the Lollipop Graph; `K_m` connected to `P_n`.
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This is the Barbell Graph without the right barbell.
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Parameters
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----------
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m, n : int or iterable container of nodes (default = 0)
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If an integer, nodes are from `range(m)` and `range(m,m+n)`.
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If a container, the entries are the coordinate of the node.
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The nodes for m appear in the complete graph $K_m$ and the nodes
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for n appear in the path $P_n$
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create_using : NetworkX graph constructor, optional (default=nx.Graph)
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Graph type to create. If graph instance, then cleared before populated.
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Notes
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-----
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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 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)
|
|
if G.is_directed():
|
|
raise NetworkXError("Directed Graph not supported")
|
|
# 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
|
|
|
|
|
|
def null_graph(create_using=None):
|
|
"""Returns 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
|
|
|
|
|
|
@nodes_or_number(0)
|
|
def path_graph(n, create_using=None):
|
|
"""Returns 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 : NetworkX graph constructor, optional (default=nx.Graph)
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
|
|
"""
|
|
n_name, nodes = n
|
|
G = empty_graph(nodes, create_using)
|
|
G.add_edges_from(pairwise(nodes))
|
|
return G
|
|
|
|
|
|
@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 : NetworkX graph constructor, optional (default=nx.Graph)
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
@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 : NetworkX graph constructor, optional (default=nx.Graph)
|
|
Graph type to create. If graph instance, then cleared before populated.
|
|
|
|
Node labels are the integers 0 to n - 1.
|
|
"""
|
|
n_name, nodes = n
|
|
if n_name == 0:
|
|
G = empty_graph(0, 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
|
|
|
|
|
|
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.
|
|
|
|
>>> 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 as e:
|
|
raise NetworkXError("Arguments must be all ints or all iterables") from e
|
|
|
|
# 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
|
|
|