Source code for networkx.algorithms.approximation.kcomponents

""" Fast approximation for k-component structure
import itertools
from collections import defaultdict
from import Mapping
from functools import cached_property

import networkx as nx
from networkx.algorithms.approximation import local_node_connectivity
from networkx.exception import NetworkXError
from networkx.utils import not_implemented_for

__all__ = ["k_components"]

[docs] @not_implemented_for("directed") @nx._dispatchable(name="approximate_k_components") def k_components(G, min_density=0.95): r"""Returns the approximate k-component structure of a graph G. A `k`-component is a maximal subgraph of a graph G that has, at least, node connectivity `k`: we need to remove at least `k` nodes to break it into more components. `k`-components have an inherent hierarchical structure because they are nested in terms of connectivity: a connected graph can contain several 2-components, each of which can contain one or more 3-components, and so forth. This implementation is based on the fast heuristics to approximate the `k`-component structure of a graph [1]_. Which, in turn, it is based on a fast approximation algorithm for finding good lower bounds of the number of node independent paths between two nodes [2]_. Parameters ---------- G : NetworkX graph Undirected graph min_density : Float Density relaxation threshold. Default value 0.95 Returns ------- k_components : dict Dictionary with connectivity level `k` as key and a list of sets of nodes that form a k-component of level `k` as values. Raises ------ NetworkXNotImplemented If G is directed. Examples -------- >>> # Petersen graph has 10 nodes and it is triconnected, thus all >>> # nodes are in a single component on all three connectivity levels >>> from networkx.algorithms import approximation as apxa >>> G = nx.petersen_graph() >>> k_components = apxa.k_components(G) Notes ----- The logic of the approximation algorithm for computing the `k`-component structure [1]_ is based on repeatedly applying simple and fast algorithms for `k`-cores and biconnected components in order to narrow down the number of pairs of nodes over which we have to compute White and Newman's approximation algorithm for finding node independent paths [2]_. More formally, this algorithm is based on Whitney's theorem, which states an inclusion relation among node connectivity, edge connectivity, and minimum degree for any graph G. This theorem implies that every `k`-component is nested inside a `k`-edge-component, which in turn, is contained in a `k`-core. Thus, this algorithm computes node independent paths among pairs of nodes in each biconnected part of each `k`-core, and repeats this procedure for each `k` from 3 to the maximal core number of a node in the input graph. Because, in practice, many nodes of the core of level `k` inside a bicomponent actually are part of a component of level k, the auxiliary graph needed for the algorithm is likely to be very dense. Thus, we use a complement graph data structure (see `AntiGraph`) to save memory. AntiGraph only stores information of the edges that are *not* present in the actual auxiliary graph. When applying algorithms to this complement graph data structure, it behaves as if it were the dense version. See also -------- k_components References ---------- .. [1] Torrents, J. and F. Ferraro (2015) Structural Cohesion: Visualization and Heuristics for Fast Computation. .. [2] White, Douglas R., and Mark Newman (2001) A Fast Algorithm for Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 .. [3] Moody, J. and D. White (2003). Social cohesion and embeddedness: A hierarchical conception of social groups. American Sociological Review 68(1), 103--28. """ # Dictionary with connectivity level (k) as keys and a list of # sets of nodes that form a k-component as values k_components = defaultdict(list) # make a few functions local for speed node_connectivity = local_node_connectivity k_core = nx.k_core core_number = nx.core_number biconnected_components = nx.biconnected_components combinations = itertools.combinations # Exact solution for k = {1,2} # There is a linear time algorithm for triconnectivity, if we had an # implementation available we could start from k = 4. for component in nx.connected_components(G): # isolated nodes have connectivity 0 comp = set(component) if len(comp) > 1: k_components[1].append(comp) for bicomponent in nx.biconnected_components(G): # avoid considering dyads as bicomponents bicomp = set(bicomponent) if len(bicomp) > 2: k_components[2].append(bicomp) # There is no k-component of k > maximum core number # \kappa(G) <= \lambda(G) <= \delta(G) g_cnumber = core_number(G) max_core = max(g_cnumber.values()) for k in range(3, max_core + 1): C = k_core(G, k, core_number=g_cnumber) for nodes in biconnected_components(C): # Build a subgraph SG induced by the nodes that are part of # each biconnected component of the k-core subgraph C. if len(nodes) < k: continue SG = G.subgraph(nodes) # Build auxiliary graph H = _AntiGraph() H.add_nodes_from(SG.nodes()) for u, v in combinations(SG, 2): K = node_connectivity(SG, u, v, cutoff=k) if k > K: H.add_edge(u, v) for h_nodes in biconnected_components(H): if len(h_nodes) <= k: continue SH = H.subgraph(h_nodes) for Gc in _cliques_heuristic(SG, SH, k, min_density): for k_nodes in biconnected_components(Gc): Gk = nx.k_core(SG.subgraph(k_nodes), k) if len(Gk) <= k: continue k_components[k].append(set(Gk)) return k_components
def _cliques_heuristic(G, H, k, min_density): h_cnumber = nx.core_number(H) for i, c_value in enumerate(sorted(set(h_cnumber.values()), reverse=True)): cands = {n for n, c in h_cnumber.items() if c == c_value} # Skip checking for overlap for the highest core value if i == 0: overlap = False else: overlap = set.intersection( *[{x for x in H[n] if x not in cands} for n in cands] ) if overlap and len(overlap) < k: SH = H.subgraph(cands | overlap) else: SH = H.subgraph(cands) sh_cnumber = nx.core_number(SH) SG = nx.k_core(G.subgraph(SH), k) while not (_same(sh_cnumber) and nx.density(SH) >= min_density): # This subgraph must be writable => .copy() SH = H.subgraph(SG).copy() if len(SH) <= k: break sh_cnumber = nx.core_number(SH) sh_deg = dict( min_deg = min(sh_deg.values()) SH.remove_nodes_from(n for n, d in sh_deg.items() if d == min_deg) SG = nx.k_core(G.subgraph(SH), k) else: yield SG def _same(measure, tol=0): vals = set(measure.values()) if (max(vals) - min(vals)) <= tol: return True return False class _AntiGraph(nx.Graph): """ Class for complement graphs. The main goal is to be able to work with big and dense graphs with a low memory footprint. In this class you add the edges that *do not exist* in the dense graph, the report methods of the class return the neighbors, the edges and the degree as if it was the dense graph. Thus it's possible to use an instance of this class with some of NetworkX functions. In this case we only use k-core, connected_components, and biconnected_components. """ all_edge_dict = {"weight": 1} def single_edge_dict(self): return self.all_edge_dict edge_attr_dict_factory = single_edge_dict # type: ignore[assignment] def __getitem__(self, n): """Returns a dict of neighbors of node n in the dense graph. Parameters ---------- n : node A node in the graph. Returns ------- adj_dict : dictionary The adjacency dictionary for nodes connected to n. """ all_edge_dict = self.all_edge_dict return { node: all_edge_dict for node in set(self._adj) - set(self._adj[n]) - {n} } def neighbors(self, n): """Returns an iterator over all neighbors of node n in the dense graph. """ try: return iter(set(self._adj) - set(self._adj[n]) - {n}) except KeyError as err: raise NetworkXError(f"The node {n} is not in the graph.") from err class AntiAtlasView(Mapping): """An adjacency inner dict for AntiGraph""" def __init__(self, graph, node): self._graph = graph self._atlas = graph._adj[node] self._node = node def __len__(self): return len(self._graph) - len(self._atlas) - 1 def __iter__(self): return (n for n in self._graph if n not in self._atlas and n != self._node) def __getitem__(self, nbr): nbrs = set(self._graph._adj) - set(self._atlas) - {self._node} if nbr in nbrs: return self._graph.all_edge_dict raise KeyError(nbr) class AntiAdjacencyView(AntiAtlasView): """An adjacency outer dict for AntiGraph""" def __init__(self, graph): self._graph = graph self._atlas = graph._adj def __len__(self): return len(self._atlas) def __iter__(self): return iter(self._graph) def __getitem__(self, node): if node not in self._graph: raise KeyError(node) return self._graph.AntiAtlasView(self._graph, node) @cached_property def adj(self): return self.AntiAdjacencyView(self) def subgraph(self, nodes): """This subgraph method returns a full AntiGraph. Not a View""" nodes = set(nodes) G = _AntiGraph() G.add_nodes_from(nodes) for n in G: Gnbrs = G.adjlist_inner_dict_factory() G._adj[n] = Gnbrs for nbr, d in self._adj[n].items(): if nbr in G._adj: Gnbrs[nbr] = d G._adj[nbr][n] = d G.graph = self.graph return G class AntiDegreeView(nx.reportviews.DegreeView): def __iter__(self): all_nodes = set(self._succ) for n in self._nodes: nbrs = all_nodes - set(self._succ[n]) - {n} yield (n, len(nbrs)) def __getitem__(self, n): nbrs = set(self._succ) - set(self._succ[n]) - {n} # AntiGraph is a ThinGraph so all edges have weight 1 return len(nbrs) + (n in nbrs) @cached_property def degree(self): """Returns an iterator for (node, degree) and degree for single node. The node degree is the number of edges adjacent to the node. Parameters ---------- nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns ------- deg: Degree of the node, if a single node is passed as argument. nd_iter : an iterator The iterator returns two-tuples of (node, degree). See Also -------- degree Examples -------- >>> G = nx.path_graph(4) >>> # node 0 with degree 1 1 >>> list([0, 1])) [(0, 1), (1, 2)] """ return self.AntiDegreeView(self) def adjacency(self): """Returns an iterator of (node, adjacency set) tuples for all nodes in the dense graph. This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included. Returns ------- adj_iter : iterator An iterator of (node, adjacency set) for all nodes in the graph. """ for n in self._adj: yield (n, set(self._adj) - set(self._adj[n]) - {n})