Source code for networkx.generators.classic

"""Generators for some classic graphs.

The typical graph builder function 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 functions
in this module return a Graph class (i.e. a simple, undirected graph).

"""

import itertools
import numbers

import networkx as nx
from networkx.classes import Graph
from networkx.exception import NetworkXError
from networkx.utils import nodes_or_number, pairwise

__all__ = [
    "balanced_tree",
    "barbell_graph",
    "binomial_tree",
    "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",
    "tadpole_graph",
    "trivial_graph",
    "turan_graph",
    "wheel_graph",
]


# -------------------------------------------------------------------
#   Some Classic Graphs
# -------------------------------------------------------------------


def _tree_edges(n, r):
    if n == 0:
        return
    # 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


[docs] @nx._dispatch(graphs=None) def full_rary_tree(r, n, create_using=None): """Creates a full r-ary tree of `n` nodes. Sometimes called a k-ary, n-ary, or m-ary tree. "... all non-leaf nodes 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 : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. 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] @nx._dispatch(graphs=None) def balanced_tree(r, h, create_using=None): """Returns 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 : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. 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] @nx._dispatch(graphs=None) def barbell_graph(m1, m2, create_using=None): """Returns the Barbell Graph: two complete graphs connected by a path. Parameters ---------- m1 : int Size of the left and right barbells, must be greater than 2. m2 : int Length of the path connecting the barbells. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Only undirected Graphs are supported. Returns ------- G : NetworkX graph A barbell graph. Notes ----- 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 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) if G.is_directed(): raise NetworkXError("Directed Graph not supported") # 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] @nx._dispatch(graphs=None) def binomial_tree(n, create_using=None): """Returns the Binomial Tree of order n. The binomial tree of order 0 consists of a single node. A binomial tree of order k is defined recursively by linking two binomial trees of order k-1: the root of one is the leftmost child of the root of the other. Parameters ---------- n : int Order of the binomial tree. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : NetworkX graph A binomial tree of $2^n$ nodes and $2^n - 1$ edges. """ G = nx.empty_graph(1, create_using) N = 1 for i in range(n): # Use G.edges() to ensure 2-tuples. G.edges is 3-tuple for MultiGraph edges = [(u + N, v + N) for (u, v) in G.edges()] G.add_edges_from(edges) G.add_edge(0, N) N *= 2 return G
[docs] @nodes_or_number(0) @nx._dispatch(graphs=None) def complete_graph(n, create_using=None): """Return the complete graph `K_n` with n nodes. A complete graph on `n` nodes means that all pairs of distinct nodes have an edge connecting them. 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. Warning: n is not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. 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 """ _, nodes = n G = empty_graph(nodes, 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] @nx._dispatch(graphs=None) def circular_ladder_graph(n, create_using=None): """Returns 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
[docs] @nx._dispatch(graphs=None) def circulant_graph(n, offsets, create_using=None): r"""Returns the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ nodes. The circulant graph $Ci_n(x_1, ..., x_m)$ consists of $n$ nodes $0, ..., n-1$ such that node $i$ is connected to nodes $(i + x) \mod n$ and $(i - x) \mod n$ for all $x$ in $x_1, ..., x_m$. Thus $Ci_n(1)$ is a cycle graph. Parameters ---------- n : integer The number of nodes in the graph. offsets : list of integers A list of node offsets, $x_1$ up to $x_m$, as described above. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- NetworkX Graph of type create_using Examples -------- Many well-known graph families are subfamilies of the circulant graphs; for example, to create the cycle graph on n points, we connect every node to nodes on either side (with offset plus or minus one). For n = 10, >>> G = nx.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 create the complete graph on 5 points with the set of offsets [1, 2]: >>> G = nx.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) @nx._dispatch(graphs=None) def cycle_graph(n, create_using=None): """Returns 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. Warning: n is not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Notes ----- If create_using is directed, the direction is in increasing order. """ _, nodes = n G = empty_graph(nodes, create_using) G.add_edges_from(pairwise(nodes, cyclic=True)) return G
[docs] @nx._dispatch(graphs=None) def dorogovtsev_goltsev_mendes_graph(n, create_using=None): """Returns the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph. The Dorogovtsev-Goltsev-Mendes [1]_ procedure produces a scale-free graph deterministically with the following properties for a given `n`: - Total number of nodes = ``3 * (3**n + 1) / 2`` - Total number of edges = ``3 ** (n + 1)`` Parameters ---------- n : integer The generation number. create_using : NetworkX Graph, optional Graph type to be returned. Directed graphs and multi graphs are not supported. Returns ------- G : NetworkX Graph Examples -------- >>> G = nx.dorogovtsev_goltsev_mendes_graph(3) >>> G.number_of_nodes() 15 >>> G.number_of_edges() 27 >>> nx.is_planar(G) True References ---------- .. [1] S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, "Pseudofractal scale-free web", Physical Review E 65, 066122, 2002. https://arxiv.org/pdf/cond-mat/0112143.pdf """ G = empty_graph(0, create_using) if G.is_directed(): raise NetworkXError("Directed Graph not supported") if G.is_multigraph(): raise NetworkXError("Multigraph not supported") 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(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) @nx._dispatch(graphs=None) def empty_graph(n=0, create_using=None, default=Graph): """Returns 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 Instance, Constructor or None Indicator of type of graph to return. If a Graph-type instance, then clear and use it. If None, use the `default` constructor. If a constructor, call it to create an empty graph. default : Graph constructor (optional, default = nx.Graph) The constructor to use if create_using is None. If None, then nx.Graph is used. This is used when passing an unknown `create_using` value through your home-grown function to `empty_graph` and you want a default constructor other than nx.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 be a Graph Constructor or a "graph"-like object. Constructors, e.g. `nx.Graph` or `nx.MultiGraph` will be used to create the returned graph. "graph"-like objects 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 three 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. Thirdly, when constructing your home-grown graph creation function you can use empty_graph to construct the graph by passing a user defined create_using to empty_graph. In this case, if you want the default constructor to be other than nx.Graph, specify `default`. >>> def mygraph(n, create_using=None): ... G = nx.empty_graph(n, create_using, nx.MultiGraph) ... G.add_edges_from([(0, 1), (0, 1)]) ... return G >>> G = mygraph(3) >>> G.is_multigraph() True >>> G = mygraph(3, nx.Graph) >>> G.is_multigraph() False See also create_empty_copy(G). """ if create_using is None: G = default() elif isinstance(create_using, type): G = create_using() elif not hasattr(create_using, "adj"): raise TypeError("create_using is not a valid NetworkX graph type or instance") else: # create_using is a NetworkX style Graph create_using.clear() G = create_using _, nodes = n G.add_nodes_from(nodes) return G
[docs] @nx._dispatch(graphs=None) def ladder_graph(n, create_using=None): """Returns 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. """ G = empty_graph(2 * n, create_using) if G.is_directed(): raise NetworkXError("Directed Graph not supported") 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]) @nx._dispatch(graphs=None) def lollipop_graph(m, n, create_using=None): """Returns 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 of nodes, those nodes appear in the graph. Warning: m and n are not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. The nodes for m appear in the complete graph $K_m$ and the nodes for n appear in the path $P_n$ create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. 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 M = len(m_nodes) if M < 2: raise NetworkXError("Invalid description: m should indicate at least 2 nodes") n, n_nodes = n if isinstance(m, numbers.Integral) and isinstance(n, numbers.Integral): n_nodes = list(range(M, M + n)) N = len(n_nodes) # 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)) if len(G) != M + N: raise NetworkXError("Nodes must be distinct in containers m and n") # connect ball to stick if M > 0 and N > 0: G.add_edge(m_nodes[-1], n_nodes[0]) return G
[docs] @nx._dispatch(graphs=None) 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
[docs] @nodes_or_number(0) @nx._dispatch(graphs=None) 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, nodes are 0 to n - 1. If an iterable of nodes, in the order they appear in the path. Warning: n is not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. """ _, nodes = n G = empty_graph(nodes, create_using) G.add_edges_from(pairwise(nodes)) return G
[docs] @nodes_or_number(0) @nx._dispatch(graphs=None) 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. Warning: n is not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. 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, nodes = n if isinstance(n, numbers.Integral): nodes.append(n) # there should be n+1 nodes G = empty_graph(nodes, create_using) if G.is_directed(): raise NetworkXError("Directed Graph not supported") if len(nodes) > 1: hub, *spokes = nodes G.add_edges_from((hub, node) for node in spokes) return G
[docs] @nodes_or_number([0, 1]) @nx._dispatch(graphs=None) def tadpole_graph(m, n, create_using=None): """Returns the (m,n)-tadpole graph; `C_m` connected to `P_n`. This graph on m+n nodes connects a cycle of size m to a path of length n. It looks like a tadpole. It is also called a kite graph or a dragon graph. 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 of nodes, those nodes appear in the graph. Warning: m and n are not checked for duplicates and if present the resulting graph may not be as desired. The nodes for m appear in the cycle graph $C_m$ and the nodes for n appear in the path $P_n$. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Raises ------- NetworkXError If `m < 2`. The tadpole graph is undefined for `m<2`. Notes ----- The 2 subgraphs are joined via an edge (m-1, m). If n=0, this is a cycle graph. m and/or n can be a container of nodes instead of an integer. """ m, m_nodes = m M = len(m_nodes) if M < 2: raise NetworkXError("Invalid description: m should indicate at least 2 nodes") n, n_nodes = n if isinstance(m, numbers.Integral) and isinstance(n, numbers.Integral): n_nodes = list(range(M, M + n)) N = len(n_nodes) # the circle G = cycle_graph(m_nodes, create_using) if G.is_directed(): raise NetworkXError("Directed Graph not supported") # the stick nx.add_path(G, [m_nodes[-1]] + list(n_nodes)) return G
[docs] @nx._dispatch(graphs=None) 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] @nx._dispatch(graphs=None) def turan_graph(n, r): r"""Return the Turan Graph The Turan Graph is a complete multipartite graph on $n$ nodes with $r$ disjoint subsets. That is, edges connect each node to every node not in its subset. Given $n$ and $r$, we create 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 nodes. 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) @nx._dispatch(graphs=None) 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. Warning: n is not checked for duplicates and if present the resulting graph may not be as desired. Make sure you have no duplicates. 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. """ _, nodes = n G = empty_graph(nodes, create_using) if G.is_directed(): raise NetworkXError("Directed Graph not supported") if len(nodes) > 1: hub, *rim = nodes G.add_edges_from((hub, node) for node in rim) if len(rim) > 1: G.add_edges_from(pairwise(rim, cyclic=True)) return G
[docs] @nx._dispatch(graphs=None) 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 nodes 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 nodes, 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 builder 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(itertools.accumulate((0,) + subset_sizes)) subsets = [range(start, end) for start, end in extents] except TypeError: subsets = subset_sizes else: if any(size < 0 for size in subset_sizes): raise NetworkXError(f"Negative number of nodes not valid: {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 err: raise NetworkXError("Arguments must be all ints or all iterables") from err # Across subsets, all nodes 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