## Mercurial > repos > shellac > sam_consensus_v3

### view env/lib/python3.9/site-packages/networkx/algorithms/minors.py @ 0:4f3585e2f14b draft default tip

Find changesets by keywords (author, files, the commit message), revision
number or hash, or revset expression.

"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"

author | shellac |
---|---|

date | Mon, 22 Mar 2021 18:12:50 +0000 |

parents | |

children |

line wrap: on

line source

"""Provides functions for computing minors of a graph.""" from itertools import chain from itertools import combinations from itertools import permutations from itertools import 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", "identified_nodes", "quotient_graph"] chaini = chain.from_iterable def equivalence_classes(iterable, relation): """Returns the set of equivalence classes of the given `iterable` under the specified equivalence relation. `relation` must be a Boolean-valued function that takes two argument. It must represent an equivalence relation (that is, the relation induced by the function must be reflexive, symmetric, and transitive). The return value is a set of sets. It is a partition of the elements of `iterable`; duplicate elements will be ignored so it makes the most sense for `iterable` to be a :class:`set`. """ # 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} def quotient_graph( G, partition, edge_relation=None, node_data=None, edge_data=None, 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 list of 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 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 `G` in the partition induced by `node_relation`. 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`. 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. 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. 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)] .. _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 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) return _quotient_graph( G, partition, edge_relation, node_data, edge_data, relabel, create_using ) # 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) return _quotient_graph( G, partition, edge_relation, node_data, edge_data, relabel, create_using ) def _quotient_graph( G, partition, edge_relation=None, node_data=None, edge_data=None, relabel=False, create_using=None, ): # Each node in the graph must be in exactly one block. if any(sum(1 for b in partition if v in b) != 1 for v in G): raise NetworkXException("each node must be in exactly one block") 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 dict( 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 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. 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)] See also -------- contracted_edge quotient_graph Notes ----- This function is also available as `identified_nodes`. """ # 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(): in_edges = ( (w if w != v else u, u, d) for w, x, d in G.in_edges(v, data=True) if self_loops or w != u ) out_edges = ( (u, w if w != v else u, d) for x, w, d in G.out_edges(v, data=True) if self_loops or w != u ) new_edges = chain(in_edges, out_edges) else: new_edges = ( (u, w if w != v else u, d) for x, w, d in G.edges(v, data=True) if self_loops or w != u ) # If the H=G, the generators change as H changes # This makes the new_edges independent of H if not copy: new_edges = list(new_edges) v_data = H.nodes[v] H.remove_node(v) H.add_edges_from(new_edges) 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 def contracted_edge(G, edge, self_loops=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. 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 """ if not G.has_edge(*edge): raise ValueError(f"Edge {edge} does not exist in graph G; cannot contract it") return contracted_nodes(G, *edge, self_loops=self_loops)