Source code for networkx.algorithms.operators.product

"""
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",
    "corona_product",
]


def _dict_product(d1, d2):
    return {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


[docs]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
[docs]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
[docs]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
[docs]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
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def power(G, k): """Returns the specified power of a graph. The $k$th power of a simple graph $G$, denoted $G^k$, is a graph on the same set of nodes in which two distinct nodes $u$ and $v$ are adjacent in $G^k$ if and only if the shortest path distance between $u$ and $v$ in $G$ is at most $k$. Parameters ---------- G : graph A NetworkX simple graph object. k : positive integer The power to which to raise the graph `G`. Returns ------- NetworkX simple graph `G` to the power `k`. Raises ------ ValueError If the exponent `k` is not positive. NetworkXNotImplemented If `G` is not a simple graph. Examples -------- The number of edges will never decrease when taking successive powers: >>> G = nx.path_graph(4) >>> list(nx.power(G, 2).edges) [(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)] >>> list(nx.power(G, 3).edges) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] The `k`th power of a cycle graph on *n* nodes is the complete graph on *n* nodes, if `k` is at least ``n // 2``: >>> G = nx.cycle_graph(5) >>> H = nx.complete_graph(5) >>> nx.is_isomorphic(nx.power(G, 2), H) True >>> G = nx.cycle_graph(8) >>> H = nx.complete_graph(8) >>> nx.is_isomorphic(nx.power(G, 4), H) True References ---------- .. [1] J. A. Bondy, U. S. R. Murty, *Graph Theory*. Springer, 2008. Notes ----- This definition of "power graph" comes from Exercise 3.1.6 of *Graph Theory* by Bondy and Murty [1]_. """ if k <= 0: raise ValueError("k must be a positive integer") H = nx.Graph() H.add_nodes_from(G) # update BFS code to ignore self loops. for n in G: seen = {} # level (number of hops) when seen in BFS level = 1 # the current level nextlevel = G[n] while nextlevel: thislevel = nextlevel # advance to next level nextlevel = {} # and start a new list (fringe) for v in thislevel: if v == n: # avoid self loop continue if v not in seen: seen[v] = level # set the level of vertex v nextlevel.update(G[v]) # add neighbors of v if k <= level: break level += 1 H.add_edges_from((n, nbr) for nbr in seen) return H
[docs]@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
[docs]@not_implemented_for("directed") @not_implemented_for("multigraph") def corona_product(G, H): r"""Returns the Corona product of G and H. The corona product of $G$ and $H$ is the graph $C = G \circ H$ obtained by taking one copy of $G$, called the center graph, $|V(G)|$ copies of $H$, called the outer graph, and making the $i$-th vertex of $G$ adjacent to every vertex of the $i$-th copy of $H$, where $1 ≤ i ≤ |V(G)|$. Parameters ---------- G, H: NetworkX graphs The graphs to take the carona product of. `G` is the center graph and `H` is the outer graph Returns ------- C: NetworkX graph The Corona product of G and H. Raises ------ NetworkXError If G and H are not both directed or both undirected. Examples -------- >>> G = nx.cycle_graph(4) >>> H = nx.path_graph(2) >>> C = nx.corona_product(G, H) >>> list(C) [0, 1, 2, 3, (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)] >>> print(C) Graph with 12 nodes and 16 edges References ---------- [1] M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi, "Studying the corona product of graphs under some graph invariants," Transactions on Combinatorics, vol. 3, no. 3, pp. 43–49, Sep. 2014, doi: 10.22108/toc.2014.5542. [2] A. Faraji, "Corona Product in Graph Theory," Ali Faraji, May 11, 2021. https://blog.alifaraji.ir/math/graph-theory/corona-product.html (accessed Dec. 07, 2021). """ GH = _init_product_graph(G, H) GH.add_nodes_from(G) GH.add_edges_from(G.edges) for G_node in G: # copy nodes of H in GH, call it H_i GH.add_nodes_from((G_node, v) for v in H) # copy edges of H_i based on H GH.add_edges_from( ((G_node, e0), (G_node, e1), d) for e0, e1, d in H.edges.data() ) # creating new edges between H_i and a G's node GH.add_edges_from((G_node, (G_node, H_node)) for H_node in H) return GH