375 lines
13 KiB
Python
375 lines
13 KiB
Python
""" Fast approximation for k-component structure
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"""
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# Copyright (C) 2015 by
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# Jordi Torrents <jtorrents@milnou.net>
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# All rights reserved.
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# BSD license.
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import itertools
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from collections import defaultdict
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from collections.abc import Mapping
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import networkx as nx
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from networkx.exception import NetworkXError
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from networkx.utils import not_implemented_for
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from networkx.algorithms.approximation import local_node_connectivity
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from networkx.algorithms.connectivity import \
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local_node_connectivity as exact_local_node_connectivity
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__author__ = """\n""".join(['Jordi Torrents <jtorrents@milnou.net>'])
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__all__ = ['k_components']
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not_implemented_for('directed')
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def k_components(G, min_density=0.95):
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r"""Returns the approximate k-component structure of a graph G.
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A `k`-component is a maximal subgraph of a graph G that has, at least,
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node connectivity `k`: we need to remove at least `k` nodes to break it
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into more components. `k`-components have an inherent hierarchical
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structure because they are nested in terms of connectivity: a connected
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graph can contain several 2-components, each of which can contain
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one or more 3-components, and so forth.
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This implementation is based on the fast heuristics to approximate
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the `k`-component structure of a graph [1]_. Which, in turn, it is based on
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a fast approximation algorithm for finding good lower bounds of the number
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of node independent paths between two nodes [2]_.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph
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min_density : Float
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Density relaxation threshold. Default value 0.95
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Returns
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-------
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k_components : dict
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Dictionary with connectivity level `k` as key and a list of
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sets of nodes that form a k-component of level `k` as values.
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Examples
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--------
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>>> # Petersen graph has 10 nodes and it is triconnected, thus all
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>>> # nodes are in a single component on all three connectivity levels
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>>> from networkx.algorithms import approximation as apxa
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>>> G = nx.petersen_graph()
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>>> k_components = apxa.k_components(G)
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Notes
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-----
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The logic of the approximation algorithm for computing the `k`-component
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structure [1]_ is based on repeatedly applying simple and fast algorithms
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for `k`-cores and biconnected components in order to narrow down the
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number of pairs of nodes over which we have to compute White and Newman's
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approximation algorithm for finding node independent paths [2]_. More
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formally, this algorithm is based on Whitney's theorem, which states
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an inclusion relation among node connectivity, edge connectivity, and
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minimum degree for any graph G. This theorem implies that every
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`k`-component is nested inside a `k`-edge-component, which in turn,
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is contained in a `k`-core. Thus, this algorithm computes node independent
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paths among pairs of nodes in each biconnected part of each `k`-core,
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and repeats this procedure for each `k` from 3 to the maximal core number
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of a node in the input graph.
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Because, in practice, many nodes of the core of level `k` inside a
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bicomponent actually are part of a component of level k, the auxiliary
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graph needed for the algorithm is likely to be very dense. Thus, we use
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a complement graph data structure (see `AntiGraph`) to save memory.
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AntiGraph only stores information of the edges that are *not* present
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in the actual auxiliary graph. When applying algorithms to this
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complement graph data structure, it behaves as if it were the dense
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version.
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See also
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--------
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k_components
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References
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----------
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.. [1] Torrents, J. and F. Ferraro (2015) Structural Cohesion:
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Visualization and Heuristics for Fast Computation.
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https://arxiv.org/pdf/1503.04476v1
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.. [2] White, Douglas R., and Mark Newman (2001) A Fast Algorithm for
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Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035
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http://eclectic.ss.uci.edu/~drwhite/working.pdf
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.. [3] Moody, J. and D. White (2003). Social cohesion and embeddedness:
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A hierarchical conception of social groups.
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American Sociological Review 68(1), 103--28.
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http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf
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"""
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# Dictionary with connectivity level (k) as keys and a list of
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# sets of nodes that form a k-component as values
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k_components = defaultdict(list)
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# make a few functions local for speed
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node_connectivity = local_node_connectivity
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k_core = nx.k_core
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core_number = nx.core_number
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biconnected_components = nx.biconnected_components
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density = nx.density
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combinations = itertools.combinations
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# Exact solution for k = {1,2}
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# There is a linear time algorithm for triconnectivity, if we had an
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# implementation available we could start from k = 4.
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for component in nx.connected_components(G):
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# isolated nodes have connectivity 0
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comp = set(component)
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if len(comp) > 1:
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k_components[1].append(comp)
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for bicomponent in nx.biconnected_components(G):
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# avoid considering dyads as bicomponents
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bicomp = set(bicomponent)
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if len(bicomp) > 2:
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k_components[2].append(bicomp)
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# There is no k-component of k > maximum core number
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# \kappa(G) <= \lambda(G) <= \delta(G)
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g_cnumber = core_number(G)
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max_core = max(g_cnumber.values())
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for k in range(3, max_core + 1):
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C = k_core(G, k, core_number=g_cnumber)
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for nodes in biconnected_components(C):
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# Build a subgraph SG induced by the nodes that are part of
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# each biconnected component of the k-core subgraph C.
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if len(nodes) < k:
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continue
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SG = G.subgraph(nodes)
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# Build auxiliary graph
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H = _AntiGraph()
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H.add_nodes_from(SG.nodes())
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for u, v in combinations(SG, 2):
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K = node_connectivity(SG, u, v, cutoff=k)
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if k > K:
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H.add_edge(u, v)
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for h_nodes in biconnected_components(H):
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if len(h_nodes) <= k:
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continue
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SH = H.subgraph(h_nodes)
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for Gc in _cliques_heuristic(SG, SH, k, min_density):
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for k_nodes in biconnected_components(Gc):
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Gk = nx.k_core(SG.subgraph(k_nodes), k)
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if len(Gk) <= k:
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continue
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k_components[k].append(set(Gk))
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return k_components
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def _cliques_heuristic(G, H, k, min_density):
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h_cnumber = nx.core_number(H)
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for i, c_value in enumerate(sorted(set(h_cnumber.values()), reverse=True)):
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cands = set(n for n, c in h_cnumber.items() if c == c_value)
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# Skip checking for overlap for the highest core value
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if i == 0:
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overlap = False
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else:
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overlap = set.intersection(*[
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set(x for x in H[n] if x not in cands)
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for n in cands])
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if overlap and len(overlap) < k:
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SH = H.subgraph(cands | overlap)
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else:
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SH = H.subgraph(cands)
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sh_cnumber = nx.core_number(SH)
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SG = nx.k_core(G.subgraph(SH), k)
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while not (_same(sh_cnumber) and nx.density(SH) >= min_density):
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#!! This subgraph must be writable => .copy()
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SH = H.subgraph(SG).copy()
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if len(SH) <= k:
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break
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sh_cnumber = nx.core_number(SH)
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sh_deg = dict(SH.degree())
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min_deg = min(sh_deg.values())
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SH.remove_nodes_from(n for n, d in sh_deg.items() if d == min_deg)
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SG = nx.k_core(G.subgraph(SH), k)
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else:
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yield SG
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def _same(measure, tol=0):
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vals = set(measure.values())
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if (max(vals) - min(vals)) <= tol:
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return True
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return False
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class _AntiGraph(nx.Graph):
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"""
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Class for complement graphs.
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The main goal is to be able to work with big and dense graphs with
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a low memory foodprint.
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In this class you add the edges that *do not exist* in the dense graph,
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the report methods of the class return the neighbors, the edges and
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the degree as if it was the dense graph. Thus it's possible to use
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an instance of this class with some of NetworkX functions. In this
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case we only use k-core, connected_components, and biconnected_components.
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"""
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all_edge_dict = {'weight': 1}
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def single_edge_dict(self):
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return self.all_edge_dict
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edge_attr_dict_factory = single_edge_dict
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def __getitem__(self, n):
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"""Returns a dict of neighbors of node n in the dense graph.
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Parameters
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----------
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n : node
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A node in the graph.
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Returns
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-------
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adj_dict : dictionary
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The adjacency dictionary for nodes connected to n.
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"""
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all_edge_dict = self.all_edge_dict
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return {node: all_edge_dict for node in
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set(self._adj) - set(self._adj[n]) - set([n])}
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def neighbors(self, n):
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"""Returns an iterator over all neighbors of node n in the
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dense graph.
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"""
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try:
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return iter(set(self._adj) - set(self._adj[n]) - set([n]))
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except KeyError:
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raise NetworkXError("The node %s is not in the graph." % (n,))
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class AntiAtlasView(Mapping):
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"""An adjacency inner dict for AntiGraph"""
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def __init__(self, graph, node):
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self._graph = graph
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self._atlas = graph._adj[node]
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self._node = node
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def __len__(self):
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return len(self._graph) - len(self._atlas) - 1
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def __iter__(self):
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return (n for n in self._graph if n not in self._atlas and n != self._node)
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def __getitem__(self, nbr):
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nbrs = set(self._graph._adj) - set(self._atlas) - set([self._node])
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if nbr in nbrs:
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return self._graph.all_edge_dict
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raise KeyError(nbr)
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class AntiAdjacencyView(AntiAtlasView):
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"""An adjacency outer dict for AntiGraph"""
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def __init__(self, graph):
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self._graph = graph
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self._atlas = graph._adj
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def __len__(self):
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return len(self._atlas)
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def __iter__(self):
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return iter(self._graph)
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def __getitem__(self, node):
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if node not in self._graph:
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raise KeyError(node)
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return self._graph.AntiAtlasView(self._graph, node)
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@property
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def adj(self):
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return self.AntiAdjacencyView(self)
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def subgraph(self, nodes):
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"""This subgraph method returns a full AntiGraph. Not a View"""
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nodes = set(nodes)
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G = _AntiGraph()
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G.add_nodes_from(nodes)
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for n in G:
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Gnbrs = G.adjlist_inner_dict_factory()
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G._adj[n] = Gnbrs
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for nbr, d in self._adj[n].items():
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if nbr in G._adj:
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Gnbrs[nbr] = d
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G._adj[nbr][n] = d
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G.graph = self.graph
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return G
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class AntiDegreeView(nx.reportviews.DegreeView):
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def __iter__(self):
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all_nodes = set(self._succ)
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for n in self._nodes:
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nbrs = all_nodes - set(self._succ[n]) - set([n])
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yield (n, len(nbrs))
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def __getitem__(self, n):
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nbrs = set(self._succ) - set(self._succ[n]) - set([n])
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# AntiGraph is a ThinGraph so all edges have weight 1
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return len(nbrs) + (n in nbrs)
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@property
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def degree(self):
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"""Returns an iterator for (node, degree) and degree for single node.
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The node degree is the number of edges adjacent to the node.
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Parameters
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----------
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nbunch : iterable container, optional (default=all nodes)
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A container of nodes. The container will be iterated
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through once.
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weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used
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as a weight. If None, then each edge has weight 1.
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The degree is the sum of the edge weights adjacent to the node.
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Returns
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-------
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deg:
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Degree of the node, if a single node is passed as argument.
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nd_iter : an iterator
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The iterator returns two-tuples of (node, degree).
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See Also
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--------
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degree
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Examples
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--------
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>>> G = nx.path_graph(4)
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>>> G.degree(0) # node 0 with degree 1
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1
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>>> list(G.degree([0,1]))
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[(0, 1), (1, 2)]
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"""
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return self.AntiDegreeView(self)
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def adjacency(self):
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"""Returns an iterator of (node, adjacency set) tuples for all nodes
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in the dense graph.
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This is the fastest way to look at every edge.
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For directed graphs, only outgoing adjacencies are included.
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Returns
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-------
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adj_iter : iterator
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An iterator of (node, adjacency set) for all nodes in
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the graph.
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"""
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for n in self._adj:
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yield (n, set(self._adj) - set(self._adj[n]) - set([n]))
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