### diff env/lib/python3.9/site-packages/networkx/algorithms/approximation/kcomponents.py @ 0:4f3585e2f14bdraftdefaulttip

author shellac Mon, 22 Mar 2021 18:12:50 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/env/lib/python3.9/site-packages/networkx/algorithms/approximation/kcomponents.py	Mon Mar 22 18:12:50 2021 +0000
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+""" Fast approximation for k-component structure
+"""
+import itertools
+from collections import defaultdict
+from collections.abc import Mapping
+
+import networkx as nx
+from networkx.exception import NetworkXError
+from networkx.utils import not_implemented_for
+
+from networkx.algorithms.approximation import local_node_connectivity
+
+
+__all__ = ["k_components"]
+
+
+not_implemented_for("directed")
+
+
+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 _. 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 _.
+
+    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.
+
+
+    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 _ 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 _. 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.
+
+    --------
+    k_components
+
+    References
+    ----------
+    ..   Torrents, J. and F. Ferraro (2015) Structural Cohesion:
+            Visualization and Heuristics for Fast Computation.
+            https://arxiv.org/pdf/1503.04476v1
+
+    ..   White, Douglas R., and Mark Newman (2001) A Fast Algorithm for
+            Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
+            http://eclectic.ss.uci.edu/~drwhite/working.pdf
+
+    ..   Moody, J. and D. White (2003). Social cohesion and embeddedness:
+            A hierarchical conception of social groups.
+            American Sociological Review 68(1), 103--28.
+            http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf
+
+    """
+    # 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
+    density = nx.density
+    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.append(comp)
+    for bicomponent in nx.biconnected_components(G):
+        # avoid considering dyads as bicomponents
+        bicomp = set(bicomponent)
+        if len(bicomp) > 2:
+            k_components.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()
+            for u, v in combinations(SG, 2):
+                K = node_connectivity(SG, u, v, cutoff=k)
+                if k > K:
+            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(SH.degree())
+            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 foodprint.
+
+    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
+
+    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
+        -------
+           The adjacency dictionary for nodes connected to n.
+
+        """
+        all_edge_dict = self.all_edge_dict
+        return {
+        }
+
+    def neighbors(self, n):
+        """Returns an iterator over all neighbors of node n in the
+           dense graph.
+        """
+        try:
+        except KeyError as e:
+            raise NetworkXError(f"The node {n} is not in the graph.") from e
+
+    class AntiAtlasView(Mapping):
+        """An adjacency inner dict for AntiGraph"""
+
+        def __init__(self, graph, node):
+            self._graph = graph
+            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)
+
+        """An adjacency outer dict for AntiGraph"""
+
+        def __init__(self, graph):
+            self._graph = graph
+
+        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)
+
+    @property
+
+    def subgraph(self, nodes):
+        """This subgraph method returns a full AntiGraph. Not a View"""
+        nodes = set(nodes)
+        G = _AntiGraph()
+        for n in G:
+            for nbr, d in self._adj[n].items():
+                    Gnbrs[nbr] = 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)
+
+    @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).
+
+        --------
+        degree
+
+        Examples
+        --------
+        >>> G = nx.path_graph(4)
+        >>> G.degree(0)  # node 0 with degree 1
+        1
+        >>> list(G.degree([0, 1]))
+        [(0, 1), (1, 2)]
+
+        """
+        return self.AntiDegreeView(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
+        -------
+            yield (n, set(self._adj) - set(self._adj[n]) - {n})