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diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/minors/contraction.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/minors/contraction.py new file mode 100644 index 00000000..e85b5778 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/minors/contraction.py @@ -0,0 +1,634 @@ +"""Provides functions for computing minors of a graph.""" + +from itertools import chain, combinations, permutations, product + +import networkx as nx +from networkx import density +from networkx.exception import NetworkXException +from networkx.utils import arbitrary_element + +__all__ = [ + "contracted_edge", + "contracted_nodes", + "equivalence_classes", + "identified_nodes", + "quotient_graph", +] + +chaini = chain.from_iterable + + +def equivalence_classes(iterable, relation): + """Returns equivalence classes of `relation` when applied to `iterable`. + + The equivalence classes, or blocks, consist of objects from `iterable` + which are all equivalent. They are defined to be equivalent if the + `relation` function returns `True` when passed any two objects from that + class, and `False` otherwise. To define an equivalence relation the + function must be reflexive, symmetric and transitive. + + Parameters + ---------- + iterable : list, tuple, or set + An iterable of elements/nodes. + + relation : function + A Boolean-valued function that implements an equivalence relation + (reflexive, symmetric, transitive binary relation) on the elements + of `iterable` - it must take two elements and return `True` if + they are related, or `False` if not. + + Returns + ------- + set of frozensets + A set of frozensets representing the partition induced by the equivalence + relation function `relation` on the elements of `iterable`. Each + member set in the return set represents an equivalence class, or + block, of the partition. + + Duplicate elements will be ignored so it makes the most sense for + `iterable` to be a :class:`set`. + + Notes + ----- + This function does not check that `relation` represents an equivalence + relation. You can check that your equivalence classes provide a partition + using `is_partition`. + + Examples + -------- + Let `X` be the set of integers from `0` to `9`, and consider an equivalence + relation `R` on `X` of congruence modulo `3`: this means that two integers + `x` and `y` in `X` are equivalent under `R` if they leave the same + remainder when divided by `3`, i.e. `(x - y) mod 3 = 0`. + + The equivalence classes of this relation are `{0, 3, 6, 9}`, `{1, 4, 7}`, + `{2, 5, 8}`: `0`, `3`, `6`, `9` are all divisible by `3` and leave zero + remainder; `1`, `4`, `7` leave remainder `1`; while `2`, `5` and `8` leave + remainder `2`. We can see this by calling `equivalence_classes` with + `X` and a function implementation of `R`. + + >>> X = set(range(10)) + >>> def mod3(x, y): + ... return (x - y) % 3 == 0 + >>> equivalence_classes(X, mod3) # doctest: +SKIP + {frozenset({1, 4, 7}), frozenset({8, 2, 5}), frozenset({0, 9, 3, 6})} + """ + # For simplicity of implementation, we initialize the return value as a + # list of lists, then convert it to a set of sets at the end of the + # function. + blocks = [] + # Determine the equivalence class for each element of the iterable. + for y in iterable: + # Each element y must be in *exactly one* equivalence class. + # + # Each block is guaranteed to be non-empty + for block in blocks: + x = arbitrary_element(block) + if relation(x, y): + block.append(y) + break + else: + # If the element y is not part of any known equivalence class, it + # must be in its own, so we create a new singleton equivalence + # class for it. + blocks.append([y]) + return {frozenset(block) for block in blocks} + + +@nx._dispatchable(edge_attrs="weight", returns_graph=True) +def quotient_graph( + G, + partition, + edge_relation=None, + node_data=None, + edge_data=None, + weight="weight", + relabel=False, + create_using=None, +): + """Returns the quotient graph of `G` under the specified equivalence + relation on nodes. + + Parameters + ---------- + G : NetworkX graph + The graph for which to return the quotient graph with the + specified node relation. + + partition : function, or dict or list of lists, tuples or sets + If a function, this function must represent an equivalence + relation on the nodes of `G`. It must take two arguments *u* + and *v* and return True exactly when *u* and *v* are in the + same equivalence class. The equivalence classes form the nodes + in the returned graph. + + If a dict of lists/tuples/sets, the keys can be any meaningful + block labels, but the values must be the block lists/tuples/sets + (one list/tuple/set per block), and the blocks must form a valid + partition of the nodes of the graph. That is, each node must be + in exactly one block of the partition. + + If a list of sets, the list must form a valid partition of + the nodes of the graph. That is, each node must be in exactly + one block of the partition. + + edge_relation : Boolean function with two arguments + This function must represent an edge relation on the *blocks* of + the `partition` of `G`. It must take two arguments, *B* and *C*, + each one a set of nodes, and return True exactly when there should be + an edge joining block *B* to block *C* in the returned graph. + + If `edge_relation` is not specified, it is assumed to be the + following relation. Block *B* is related to block *C* if and + only if some node in *B* is adjacent to some node in *C*, + according to the edge set of `G`. + + node_data : function + This function takes one argument, *B*, a set of nodes in `G`, + and must return a dictionary representing the node data + attributes to set on the node representing *B* in the quotient graph. + If None, the following node attributes will be set: + + * 'graph', the subgraph of the graph `G` that this block + represents, + * 'nnodes', the number of nodes in this block, + * 'nedges', the number of edges within this block, + * 'density', the density of the subgraph of `G` that this + block represents. + + edge_data : function + This function takes two arguments, *B* and *C*, each one a set + of nodes, and must return a dictionary representing the edge + data attributes to set on the edge joining *B* and *C*, should + there be an edge joining *B* and *C* in the quotient graph (if + no such edge occurs in the quotient graph as determined by + `edge_relation`, then the output of this function is ignored). + + If the quotient graph would be a multigraph, this function is + not applied, since the edge data from each edge in the graph + `G` appears in the edges of the quotient graph. + + weight : string or None, optional (default="weight") + The name of an edge attribute that holds the numerical value + used as a weight. If None then each edge has weight 1. + + relabel : bool + If True, relabel the nodes of the quotient graph to be + nonnegative integers. Otherwise, the nodes are identified with + :class:`frozenset` instances representing the blocks given in + `partition`. + + create_using : NetworkX graph constructor, optional (default=nx.Graph) + Graph type to create. If graph instance, then cleared before populated. + + Returns + ------- + NetworkX graph + The quotient graph of `G` under the equivalence relation + specified by `partition`. If the partition were given as a + list of :class:`set` instances and `relabel` is False, + each node will be a :class:`frozenset` corresponding to the same + :class:`set`. + + Raises + ------ + NetworkXException + If the given partition is not a valid partition of the nodes of + `G`. + + Examples + -------- + The quotient graph of the complete bipartite graph under the "same + neighbors" equivalence relation is `K_2`. Under this relation, two nodes + are equivalent if they are not adjacent but have the same neighbor set. + + >>> G = nx.complete_bipartite_graph(2, 3) + >>> same_neighbors = lambda u, v: (u not in G[v] and v not in G[u] and G[u] == G[v]) + >>> Q = nx.quotient_graph(G, same_neighbors) + >>> K2 = nx.complete_graph(2) + >>> nx.is_isomorphic(Q, K2) + True + + The quotient graph of a directed graph under the "same strongly connected + component" equivalence relation is the condensation of the graph (see + :func:`condensation`). This example comes from the Wikipedia article + *`Strongly connected component`_*. + + >>> G = nx.DiGraph() + >>> edges = [ + ... "ab", + ... "be", + ... "bf", + ... "bc", + ... "cg", + ... "cd", + ... "dc", + ... "dh", + ... "ea", + ... "ef", + ... "fg", + ... "gf", + ... "hd", + ... "hf", + ... ] + >>> G.add_edges_from(tuple(x) for x in edges) + >>> components = list(nx.strongly_connected_components(G)) + >>> sorted(sorted(component) for component in components) + [['a', 'b', 'e'], ['c', 'd', 'h'], ['f', 'g']] + >>> + >>> C = nx.condensation(G, components) + >>> component_of = C.graph["mapping"] + >>> same_component = lambda u, v: component_of[u] == component_of[v] + >>> Q = nx.quotient_graph(G, same_component) + >>> nx.is_isomorphic(C, Q) + True + + Node identification can be represented as the quotient of a graph under the + equivalence relation that places the two nodes in one block and each other + node in its own singleton block. + + >>> K24 = nx.complete_bipartite_graph(2, 4) + >>> K34 = nx.complete_bipartite_graph(3, 4) + >>> C = nx.contracted_nodes(K34, 1, 2) + >>> nodes = {1, 2} + >>> is_contracted = lambda u, v: u in nodes and v in nodes + >>> Q = nx.quotient_graph(K34, is_contracted) + >>> nx.is_isomorphic(Q, C) + True + >>> nx.is_isomorphic(Q, K24) + True + + The blockmodeling technique described in [1]_ can be implemented as a + quotient graph. + + >>> G = nx.path_graph(6) + >>> partition = [{0, 1}, {2, 3}, {4, 5}] + >>> M = nx.quotient_graph(G, partition, relabel=True) + >>> list(M.edges()) + [(0, 1), (1, 2)] + + Here is the sample example but using partition as a dict of block sets. + + >>> G = nx.path_graph(6) + >>> partition = {0: {0, 1}, 2: {2, 3}, 4: {4, 5}} + >>> M = nx.quotient_graph(G, partition, relabel=True) + >>> list(M.edges()) + [(0, 1), (1, 2)] + + Partitions can be represented in various ways: + + 0. a list/tuple/set of block lists/tuples/sets + 1. a dict with block labels as keys and blocks lists/tuples/sets as values + 2. a dict with block lists/tuples/sets as keys and block labels as values + 3. a function from nodes in the original iterable to block labels + 4. an equivalence relation function on the target iterable + + As `quotient_graph` is designed to accept partitions represented as (0), (1) or + (4) only, the `equivalence_classes` function can be used to get the partitions + in the right form, in order to call `quotient_graph`. + + .. _Strongly connected component: https://en.wikipedia.org/wiki/Strongly_connected_component + + References + ---------- + .. [1] Patrick Doreian, Vladimir Batagelj, and Anuska Ferligoj. + *Generalized Blockmodeling*. + Cambridge University Press, 2004. + + """ + # If the user provided an equivalence relation as a function to compute + # the blocks of the partition on the nodes of G induced by the + # equivalence relation. + if callable(partition): + # equivalence_classes always return partition of whole G. + partition = equivalence_classes(G, partition) + if not nx.community.is_partition(G, partition): + raise nx.NetworkXException( + "Input `partition` is not an equivalence relation for nodes of G" + ) + return _quotient_graph( + G, + partition, + edge_relation, + node_data, + edge_data, + weight, + relabel, + create_using, + ) + + # If the partition is a dict, it is assumed to be one where the keys are + # user-defined block labels, and values are block lists, tuples or sets. + if isinstance(partition, dict): + partition = list(partition.values()) + + # If the user provided partition as a collection of sets. Then we + # need to check if partition covers all of G nodes. If the answer + # is 'No' then we need to prepare suitable subgraph view. + partition_nodes = set().union(*partition) + if len(partition_nodes) != len(G): + G = G.subgraph(partition_nodes) + # Each node in the graph/subgraph must be in exactly one block. + if not nx.community.is_partition(G, partition): + raise NetworkXException("each node must be in exactly one part of `partition`") + return _quotient_graph( + G, + partition, + edge_relation, + node_data, + edge_data, + weight, + relabel, + create_using, + ) + + +def _quotient_graph( + G, partition, edge_relation, node_data, edge_data, weight, relabel, create_using +): + """Construct the quotient graph assuming input has been checked""" + if create_using is None: + H = G.__class__() + else: + H = nx.empty_graph(0, create_using) + # By default set some basic information about the subgraph that each block + # represents on the nodes in the quotient graph. + if node_data is None: + + def node_data(b): + S = G.subgraph(b) + return { + "graph": S, + "nnodes": len(S), + "nedges": S.number_of_edges(), + "density": density(S), + } + + # Each block of the partition becomes a node in the quotient graph. + partition = [frozenset(b) for b in partition] + H.add_nodes_from((b, node_data(b)) for b in partition) + # By default, the edge relation is the relation defined as follows. B is + # adjacent to C if a node in B is adjacent to a node in C, according to the + # edge set of G. + # + # This is not a particularly efficient implementation of this relation: + # there are O(n^2) pairs to check and each check may require O(log n) time + # (to check set membership). This can certainly be parallelized. + if edge_relation is None: + + def edge_relation(b, c): + return any(v in G[u] for u, v in product(b, c)) + + # By default, sum the weights of the edges joining pairs of nodes across + # blocks to get the weight of the edge joining those two blocks. + if edge_data is None: + + def edge_data(b, c): + edgedata = ( + d + for u, v, d in G.edges(b | c, data=True) + if (u in b and v in c) or (u in c and v in b) + ) + return {"weight": sum(d.get(weight, 1) for d in edgedata)} + + block_pairs = permutations(H, 2) if H.is_directed() else combinations(H, 2) + # In a multigraph, add one edge in the quotient graph for each edge + # in the original graph. + if H.is_multigraph(): + edges = chaini( + ( + (b, c, G.get_edge_data(u, v, default={})) + for u, v in product(b, c) + if v in G[u] + ) + for b, c in block_pairs + if edge_relation(b, c) + ) + # In a simple graph, apply the edge data function to each pair of + # blocks to determine the edge data attributes to apply to each edge + # in the quotient graph. + else: + edges = ( + (b, c, edge_data(b, c)) for (b, c) in block_pairs if edge_relation(b, c) + ) + H.add_edges_from(edges) + # If requested by the user, relabel the nodes to be integers, + # numbered in increasing order from zero in the same order as the + # iteration order of `partition`. + if relabel: + # Can't use nx.convert_node_labels_to_integers() here since we + # want the order of iteration to be the same for backward + # compatibility with the nx.blockmodel() function. + labels = {b: i for i, b in enumerate(partition)} + H = nx.relabel_nodes(H, labels) + return H + + +@nx._dispatchable( + preserve_all_attrs=True, mutates_input={"not copy": 4}, returns_graph=True +) +def contracted_nodes(G, u, v, self_loops=True, copy=True): + """Returns the graph that results from contracting `u` and `v`. + + Node contraction identifies the two nodes as a single node incident to any + edge that was incident to the original two nodes. + + Parameters + ---------- + G : NetworkX graph + The graph whose nodes will be contracted. + + u, v : nodes + Must be nodes in `G`. + + self_loops : Boolean + If this is True, any edges joining `u` and `v` in `G` become + self-loops on the new node in the returned graph. + + copy : Boolean + If this is True (default True), make a copy of + `G` and return that instead of directly changing `G`. + + + Returns + ------- + Networkx graph + If Copy is True, + A new graph object of the same type as `G` (leaving `G` unmodified) + with `u` and `v` identified in a single node. The right node `v` + will be merged into the node `u`, so only `u` will appear in the + returned graph. + If copy is False, + Modifies `G` with `u` and `v` identified in a single node. + The right node `v` will be merged into the node `u`, so + only `u` will appear in the returned graph. + + Notes + ----- + For multigraphs, the edge keys for the realigned edges may + not be the same as the edge keys for the old edges. This is + natural because edge keys are unique only within each pair of nodes. + + For non-multigraphs where `u` and `v` are adjacent to a third node + `w`, the edge (`v`, `w`) will be contracted into the edge (`u`, + `w`) with its attributes stored into a "contraction" attribute. + + This function is also available as `identified_nodes`. + + Examples + -------- + Contracting two nonadjacent nodes of the cycle graph on four nodes `C_4` + yields the path graph (ignoring parallel edges): + + >>> G = nx.cycle_graph(4) + >>> M = nx.contracted_nodes(G, 1, 3) + >>> P3 = nx.path_graph(3) + >>> nx.is_isomorphic(M, P3) + True + + >>> G = nx.MultiGraph(P3) + >>> M = nx.contracted_nodes(G, 0, 2) + >>> M.edges + MultiEdgeView([(0, 1, 0), (0, 1, 1)]) + + >>> G = nx.Graph([(1, 2), (2, 2)]) + >>> H = nx.contracted_nodes(G, 1, 2, self_loops=False) + >>> list(H.nodes()) + [1] + >>> list(H.edges()) + [(1, 1)] + + In a ``MultiDiGraph`` with a self loop, the in and out edges will + be treated separately as edges, so while contracting a node which + has a self loop the contraction will add multiple edges: + + >>> G = nx.MultiDiGraph([(1, 2), (2, 2)]) + >>> H = nx.contracted_nodes(G, 1, 2) + >>> list(H.edges()) # edge 1->2, 2->2, 2<-2 from the original Graph G + [(1, 1), (1, 1), (1, 1)] + >>> H = nx.contracted_nodes(G, 1, 2, self_loops=False) + >>> list(H.edges()) # edge 2->2, 2<-2 from the original Graph G + [(1, 1), (1, 1)] + + See Also + -------- + contracted_edge + quotient_graph + + """ + # Copying has significant overhead and can be disabled if needed + if copy: + H = G.copy() + else: + H = G + + # edge code uses G.edges(v) instead of G.adj[v] to handle multiedges + if H.is_directed(): + edges_to_remap = chain(G.in_edges(v, data=True), G.out_edges(v, data=True)) + else: + edges_to_remap = G.edges(v, data=True) + + # If the H=G, the generators change as H changes + # This makes the edges_to_remap independent of H + if not copy: + edges_to_remap = list(edges_to_remap) + + v_data = H.nodes[v] + H.remove_node(v) + + for prev_w, prev_x, d in edges_to_remap: + w = prev_w if prev_w != v else u + x = prev_x if prev_x != v else u + + if ({prev_w, prev_x} == {u, v}) and not self_loops: + continue + + if not H.has_edge(w, x) or G.is_multigraph(): + H.add_edge(w, x, **d) + else: + if "contraction" in H.edges[(w, x)]: + H.edges[(w, x)]["contraction"][(prev_w, prev_x)] = d + else: + H.edges[(w, x)]["contraction"] = {(prev_w, prev_x): d} + + if "contraction" in H.nodes[u]: + H.nodes[u]["contraction"][v] = v_data + else: + H.nodes[u]["contraction"] = {v: v_data} + return H + + +identified_nodes = contracted_nodes + + +@nx._dispatchable( + preserve_edge_attrs=True, mutates_input={"not copy": 3}, returns_graph=True +) +def contracted_edge(G, edge, self_loops=True, copy=True): + """Returns the graph that results from contracting the specified edge. + + Edge contraction identifies the two endpoints of the edge as a single node + incident to any edge that was incident to the original two nodes. A graph + that results from edge contraction is called a *minor* of the original + graph. + + Parameters + ---------- + G : NetworkX graph + The graph whose edge will be contracted. + + edge : tuple + Must be a pair of nodes in `G`. + + self_loops : Boolean + If this is True, any edges (including `edge`) joining the + endpoints of `edge` in `G` become self-loops on the new node in the + returned graph. + + copy : Boolean (default True) + If this is True, a the contraction will be performed on a copy of `G`, + otherwise the contraction will happen in place. + + Returns + ------- + Networkx graph + A new graph object of the same type as `G` (leaving `G` unmodified) + with endpoints of `edge` identified in a single node. The right node + of `edge` will be merged into the left one, so only the left one will + appear in the returned graph. + + Raises + ------ + ValueError + If `edge` is not an edge in `G`. + + Examples + -------- + Attempting to contract two nonadjacent nodes yields an error: + + >>> G = nx.cycle_graph(4) + >>> nx.contracted_edge(G, (1, 3)) + Traceback (most recent call last): + ... + ValueError: Edge (1, 3) does not exist in graph G; cannot contract it + + Contracting two adjacent nodes in the cycle graph on *n* nodes yields the + cycle graph on *n - 1* nodes: + + >>> C5 = nx.cycle_graph(5) + >>> C4 = nx.cycle_graph(4) + >>> M = nx.contracted_edge(C5, (0, 1), self_loops=False) + >>> nx.is_isomorphic(M, C4) + True + + See also + -------- + contracted_nodes + quotient_graph + + """ + u, v = edge[:2] + if not G.has_edge(u, v): + raise ValueError(f"Edge {edge} does not exist in graph G; cannot contract it") + return contracted_nodes(G, u, v, self_loops=self_loops, copy=copy) |