""" Fast approximation for k-component structure
"""
# Copyright (C) 2015 by
# Jordi Torrents <jtorrents@milnou.net>
# All rights reserved.
# BSD license.
import itertools
from collections import defaultdict, 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
from networkx.algorithms.connectivity import \
local_node_connectivity as exact_local_node_connectivity
__author__ = """\n""".join(['Jordi Torrents <jtorrents@milnou.net>'])
__all__ = ['k_components']
not_implemented_for('directed')
[docs]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.
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.
https://arxiv.org/pdf/1503.04476v1
.. [2] 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
.. [3] 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[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 = set(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(*[
set(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):
"""Return 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]) - set([n])}
def neighbors(self, n):
"""Return an iterator over all neighbors of node n in the
dense graph.
"""
try:
return iter(set(self._adj) - set(self._adj[n]) - set([n]))
except KeyError:
raise NetworkXError("The node %s is not in the graph." % (n,))
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) - set([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)
@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]) - set([n])
yield (n, len(nbrs))
def __getitem__(self, n):
nbrs = set(self._succ) - set(self._succ[n]) - set([n])
# AntiGraph is a ThinGraph so all edges have weight 1
return len(nbrs) + (n in nbrs)
@property
def degree(self):
"""Return 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)
>>> G.degree(0) # node 0 with degree 1
1
>>> list(G.degree([0,1]))
[(0, 1), (1, 2)]
"""
return self.AntiDegreeView(self)
def adjacency(self):
"""Return 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]) - set([n]))