def test_overlapping_K5():
G = nx.Graph()
G.add_edges_from(combinations(range(5), 2)) # Add a five clique
G.add_edges_from(combinations(range(2, 7), 2)) # Add another five clique
c = list(k_clique_communities(G, 4))
assert_equal(c, [frozenset(range(7))])
c = set(k_clique_communities(G, 5))
assert_equal(c, set([frozenset(range(5)), frozenset(range(2, 7))]))
def _dispersion(G_u, u, v):
"""dispersion for all nodes 'v' in a ego network G_u of node 'u'"""
u_nbrs = set(G_u[u])
ST = set(n for n in G_u[v] if n in u_nbrs)
set_uv = set([u, v])
# all possible ties of connections that u and b share
possib = combinations(ST, 2)
total = 0
for (s, t) in possib:
# neighbors of s that are in G_u, not including u and v
nbrs_s = u_nbrs.intersection(G_u[s]) - set_uv
# s and t are not directly connected
if t not in nbrs_s:
# s and t do not share a connection
if nbrs_s.isdisjoint(G_u[t]):
# tick for disp(u, v)
total += 1
# neighbors that u and v share
embededness = len(ST)
if normalized:
if embededness + c != 0:
norm_disp = ((total + b)**alpha) / (embededness + c)
else:
norm_disp = (total + b)**alpha
dispersion = norm_disp
else:
dispersion = total
return dispersion
def random_tournament(n, seed=None):
r"""Returns a random tournament graph on `n` nodes.
Parameters
----------
n : int
The number of nodes in the returned graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
bool
Whether the given graph is a tournament graph.
Notes
-----
This algorithm adds, for each pair of distinct nodes, an edge with
uniformly random orientation. In other words, `\binom{n}{2}` flips
of an unbiased coin decide the orientations of the edges in the
graph.
"""
# Flip an unbiased coin for each pair of distinct nodes.
coins = (seed.random() for i in range((n * (n - 1)) // 2))
pairs = combinations(range(n), 2)
edges = ((u, v) if r < 0.5 else (v, u) for (u, v), r in zip(pairs, coins))
return nx.DiGraph(edges)
def is_matching(G, matching):
"""Decides whether the given set or dictionary represents a valid
matching in ``G``.
A *matching* in a graph is a set of edges in which no two distinct
edges share a common endpoint.
Parameters
----------
G : NetworkX graph
matching : dict or set
A dictionary or set representing a matching. If a dictionary, it
must have ``matching[u] == v`` and ``matching[v] == u`` for each
edge ``(u, v)`` in the matching. If a set, it must have elements
of the form ``(u, v)``, where ``(u, v)`` is an edge in the
matching.
Returns
-------
bool
Whether the given set or dictionary represents a valid matching
in the graph.
"""
if isinstance(matching, dict):
matching = matching_dict_to_set(matching)
# TODO This is parallelizable.
return all(len(set(e1) & set(e2)) == 0
for e1, e2 in combinations(matching, 2))
def is_tournament(G):
"""Returns True if and only if `G` is a tournament.
A tournament is a directed graph, with neither self-loops nor
multi-edges, in which there is exactly one directed edge joining
each pair of distinct nodes.
Parameters
----------
G : NetworkX graph
A directed graph representing a tournament.
Returns
-------
bool
Whether the given graph is a tournament graph.
Notes
-----
Some definitions require a self-loop on each node, but that is not
the convention used here.
"""
# In a tournament, there is exactly one directed edge joining each pair.
return (all((v in G[u]) ^ (u in G[v]) for u, v in combinations(G, 2))
and nx.number_of_selfloops(G) == 0)
def test_tree_all_pairs_lowest_common_ancestor3(self):
"""Specifying no pairs same as specifying all."""
all_pairs = chain(combinations(self.DG, 2),
((node, node) for node in self.DG))
ans = dict(tree_all_pairs_lca(self.DG, 0, all_pairs))
self.assert_has_same_pairs(ans, self.ans)
def test_several_communities(self):
# This graph is the disjoint union of five triangles.
ground_truth = set(
[frozenset(range(3 * i, 3 * (i + 1))) for i in range(5)])
edges = chain.from_iterable(combinations(c, 2) for c in ground_truth)
G = nx.Graph(edges)
self._check_communities(G, ground_truth)
def test_default_flow_function_karate_club_graph(self):
G = nx.karate_club_graph()
nx.set_edge_attributes(G, 1, 'capacity')
T = nx.gomory_hu_tree(G)
assert_true(nx.is_tree(T))
for u, v in combinations(G, 2):
cut_value, edge = self.minimum_edge_weight(T, u, v)
assert_equal(nx.minimum_cut_value(G, u, v), cut_value)
def _find_partition(G, starting_cell):
""" Find a partition of the vertices of G into cells of complete graphs
Parameters
----------
G : NetworkX Graph
starting_cell : tuple of vertices in G which form a cell
Returns
-------
List of tuples of vertices of G
Raises
------
NetworkXError
If a cell is not a complete subgraph then G is not a line graph
"""
G_partition = G.copy()
P = [starting_cell] # partition set
G_partition.remove_edges_from(list(combinations(starting_cell, 2)))
# keep list of partitioned nodes which might have an edge in G_partition
partitioned_vertices = list(starting_cell)
while G_partition.number_of_edges() > 0:
# there are still edges left and so more cells to be made
u = partitioned_vertices[-1]
deg_u = len(G_partition[u])
if deg_u == 0:
# if u has no edges left in G_partition then we have found
# all of its cells so we do not need to keep looking
partitioned_vertices.pop()
else:
# if u still has edges then we need to find its other cell
# this other cell must be a complete subgraph or else G is
# not a line graph
new_cell = [u] + list(G_partition.neighbors(u))
for u in new_cell:
for v in new_cell:
if (u != v) and (v not in G.neighbors(u)):
msg = "G is not a line graph" \
"(partition cell not a complete subgraph)"
raise nx.NetworkXError(msg)
P.append(tuple(new_cell))
G_partition.remove_edges_from(list(combinations(new_cell, 2)))
partitioned_vertices += new_cell
return P
def test_florentine_families_graph(self):
G = nx.florentine_families_graph()
nx.set_edge_attributes(G, 1, 'capacity')
for flow_func in flow_funcs:
T = nx.gomory_hu_tree(G, flow_func=flow_func)
assert_true(nx.is_tree(T))
for u, v in combinations(G, 2):
cut_value, edge = self.minimum_edge_weight(T, u, v)
assert_equal(nx.minimum_cut_value(G, u, v), cut_value)
def test_theta(self):
"""Tests that pairs of vertices adjacent if and only if their sum
weights exceeds the threshold parameter theta.
"""
G = nx.thresholded_random_geometric_graph(50, 0.25, 0.1)
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(
(G.nodes[u]['weight'] + G.nodes[v]['weight']) >= 0.1)
def _slow_edges(G, radius, p):
"""Returns edge list of node pairs within `radius` of each other
using Minkowski distance metric `p`
Works without scipy, but in `O(n^2)` time.
"""
# TODO This can be parallelized.
edges = []
for (u, pu), (v, pv) in combinations(G.nodes(data='pos'), 2):
if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius ** p:
edges.append((u, v))
return edges
def test_metric(self):
"""Tests for providing an alternate distance metric to the
generator.
"""
# Use the L1 metric.
dist = l1dist
G = nx.geographical_threshold_graph(50, 10, metric=dist)
for u, v in combinations(G, 2):
# Adjacent vertices must exceed the threshold.
if v in G[u]:
assert_true(join(G, u, v, 10, -2, dist))
# Nonadjacent vertices must not exceed the threshold.
else:
assert_false(join(G, u, v, 10, -2, dist))
def test_p(self):
"""Tests for providing an alternate distance metric to the
generator.
"""
# Use the L1 metric.
def dist(x, y):
return sum(abs(a - b) for a, b in zip(x, y))
G = nx.thresholded_random_geometric_graph(50, 0.25, 0.1, p=1)
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def test_distances(self):
"""Tests that pairs of vertices adjacent if and only if they are
within the prescribed radius.
"""
# Use the Euclidean metric, the default according to the
# documentation.
def dist(x, y):
return sqrt(sum((a - b)**2 for a, b in zip(x, y)))
G = nx.soft_random_geometric_graph(50, 0.25)
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def test_distances(self):
"""Tests that pairs of vertices adjacent if and only if their
distances meet the given threshold.
"""
# Use the Euclidean metric and alpha = -2
# the default according to the documentation.
dist = euclidean
G = nx.geographical_threshold_graph(50, 10)
for u, v in combinations(G, 2):
# Adjacent vertices must exceed the threshold.
if v in G[u]:
assert_true(join(G, u, v, 10, -2, dist))
# Nonadjacent vertices must not exceed the threshold.
else:
assert_false(join(G, u, v, 10, -2, dist))
def test_p(self):
"""Tests for providing an alternate distance metric to the
generator.
"""
# Use the L1 metric.
dist = l1dist
G = nx.random_geometric_graph(50, 0.25, p=1)
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
# Nonadjacent vertices must be at greater distance.
else:
assert_false(
dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def test_node_names(self):
"""Tests using values other than sequential numbers as node IDs.
"""
import string
nodes = list(string.ascii_lowercase)
G = nx.thresholded_random_geometric_graph(nodes, 0.25, 0.1)
assert_equal(len(G), len(nodes))
def dist(x, y):
return sqrt(sum((a - b)**2 for a, b in zip(x, y)))
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def _node_redundancy(G, v):
"""Returns the redundancy of the node `v` in the bipartite graph `G`.
If `G` is a graph with `n` nodes, the redundancy of a node is the ratio
of the "overlap" of `v` to the maximum possible overlap of `v`
according to its degree. The overlap of `v` is the number of pairs of
neighbors that have mutual neighbors themselves, other than `v`.
`v` must have at least two neighbors in `G`.
"""
n = len(G[v])
# TODO On Python 3, we could just use `G[u].keys() & G[w].keys()` instead
# of instantiating the entire sets.
overlap = sum(1 for (u, w) in combinations(G[v], 2)
if (set(G[u]) & set(G[w])) - set([v]))
return (2 * overlap) / (n * (n - 1))
def test_distances(self):
"""Tests that pairs of vertices adjacent if and only if they are
within the prescribed radius.
"""
# Use the Euclidean metric, the default according to the
# documentation.
dist = euclidean
G = nx.random_geometric_graph(50, 0.25)
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
# Nonadjacent vertices must be at greater distance.
else:
assert_false(
dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def phase3(self):
# build potential remaining edges and choose with rejection sampling
potential_edges = combinations(self.remaining_degree, 2)
# build auxiliary graph of potential edges not already in graph
H = nx.Graph([(u, v) for (u, v) in potential_edges
if not self.graph.has_edge(u, v)])
rng = self.rng
while self.remaining_degree:
if not self.suitable_edge():
raise nx.NetworkXUnfeasible('no suitable edges left')
while True:
u, v = sorted(rng.choice(list(H.edges())))
if rng.random() < self.q(u, v):
break
if rng.random() < self.p(u, v): # accept edge
self.graph.add_edge(u, v)
self.update_remaining(u, v, aux_graph=H)
def test_node_names(self):
"""Tests using values other than sequential numbers as node IDs.
"""
import string
nodes = list(string.ascii_lowercase)
G = nx.random_geometric_graph(nodes, 0.25)
assert_equal(len(G), len(nodes))
dist = euclidean
for u, v in combinations(G, 2):
# Adjacent vertices must be within the given distance.
if v in G[u]:
assert_true(dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
# Nonadjacent vertices must be at greater distance.
else:
assert_false(
dist(G.nodes[u]['pos'], G.nodes[v]['pos']) <= 0.25)
def make_max_clique_graph(G, create_using=None):
"""Returns the maximal clique graph of the given graph.
The nodes of the maximal clique graph of `G` are the cliques of
`G` and an edge joins two cliques if the cliques are not disjoint.
Parameters
----------
G : NetworkX graph
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
A graph whose nodes are the cliques of `G` and whose edges
join two cliques if they are not disjoint.
Notes
-----
This function behaves like the following code::
import networkx as nx
G = nx.make_clique_bipartite(G)
cliques = [v for v in G.nodes() if G.nodes[v]['bipartite'] == 0]
G = nx.bipartite.project(G, cliques)
G = nx.relabel_nodes(G, {-v: v - 1 for v in G})
It should be faster, though, since it skips all the intermediate
steps.
"""
if create_using is None:
B = G.__class__()
else:
B = nx.empty_graph(0, create_using)
cliques = list(enumerate(set(c) for c in find_cliques(G)))
# Add a numbered node for each clique.
B.add_nodes_from(i for i, c in cliques)
# Join cliques by an edge if they share a node.
clique_pairs = combinations(cliques, 2)
B.add_edges_from((i, j) for (i, c1), (j, c2) in clique_pairs if c1 & c2)
return B
def _consolidate(sets, k):
"""Merge sets that share k or more elements.
See: http://rosettacode.org/wiki/Set_consolidation
The iterative python implementation posted there is
faster than this because of the overhead of building a
Graph and calling nx.connected_components, but it's not
clear for us if we can use it in NetworkX because there
is no licence for the code.
"""
G = nx.Graph()
nodes = dict((i, s) for i, s in enumerate(sets))
G.add_nodes_from(nodes)
G.add_edges_from((u, v) for u, v in combinations(nodes, 2)
if len(nodes[u] & nodes[v]) >= k)
for component in nx.connected_components(G):
yield set.union(*[nodes[n] for n in component])
def _odd_triangle(G, T):
""" Test whether T is an odd triangle in G
Parameters
----------
G : NetworkX Graph
T : 3-tuple of vertices forming triangle in G
Returns
-------
True is T is an odd triangle
False otherwise
Raises
------
NetworkXError
T is not a triangle in G
Notes
-----
An odd triangle is one in which there exists another vertex in G which is
adjacent to either exactly one or exactly all three of the vertices in the
triangle.
"""
for u in T:
if u not in G.nodes():
raise nx.NetworkXError("Vertex %s not in graph" % u)
for e in list(combinations(T, 2)):
if e[0] not in G.neighbors(e[1]):
raise nx.NetworkXError("Edge (%s, %s) not in graph" % (e[0], e[1]))
T_neighbors = defaultdict(int)
for t in T:
for v in G.neighbors(t):
if v not in T:
T_neighbors[v] += 1
for v in T_neighbors:
if T_neighbors[v] in [1, 3]:
return True
return False
def test_wikipedia_example(self):
# Example from https://en.wikipedia.org/wiki/Gomory%E2%80%93Hu_tree
G = nx.Graph()
G.add_weighted_edges_from((
(0, 1, 1),
(0, 2, 7),
(1, 2, 1),
(1, 3, 3),
(1, 4, 2),
(2, 4, 4),
(3, 4, 1),
(3, 5, 6),
(4, 5, 2),
))
for flow_func in flow_funcs:
T = nx.gomory_hu_tree(G, capacity='weight', flow_func=flow_func)
assert_true(nx.is_tree(T))
for u, v in combinations(G, 2):
cut_value, edge = self.minimum_edge_weight(T, u, v)
assert_equal(nx.minimum_cut_value(G, u, v, capacity='weight'),
cut_value)
def thresholded_random_geometric_graph(n, radius, theta, dim=2,
pos=None, weight=None, p=2, seed=None):
"""Returns a thresholded random geometric graph in the unit cube.
The thresholded random geometric graph [1] model places `n` nodes
uniformly at random in the unit cube of dimensions `dim`. Each node
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
joined by an edge if they are within the maximum connection distance,
`radius` computed by the `p`-Minkowski distance and the summation of
weights :math:`w_u` + :math:`w_v` is greater than or equal
to the threshold parameter `theta`.
Edges within `radius` of each other are determined using a KDTree when
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
to :math:`O(n)`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
radius: float
Distance threshold value
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
weight : dict, optional
Node weights as a dictionary of numbers keyed by node.
p : float, optional
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the :math:`L^2` metric
(the Euclidean distance metric), p = 2 is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
Graph
A thresholded random geographic graph, undirected and without
self-loops.
Each node has a node attribute ``'pos'`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a nodethre
attribute ``'weight'`` that stores the weight of that node as
provided or as generated.
Examples
--------
Default Graph:
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
Custom Graph:
Create a thresholded random geometric graph on 50 uniformly distributed
nodes where nodes are joined by an edge if their sum weights drawn from
a exponential distribution with rate = 5 are >= theta = 0.1 and their
Euclidean distance is at most 0.2.
Notes
-----
This uses a *k*-d tree to build the graph.
The `pos` keyword argument can be used to specify node positions so you
can create an arbitrary distribution and domain for positions.
For example, to use a 2D Gaussian distribution of node positions with mean
(0, 0) and standard deviation 2
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter :math:`\lambda=1`.
To specify weights from a different distribution, use the `weight` keyword
argument::
::
>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
References
----------
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
"""
n_name, nodes = n
G = nx.Graph()
namestr = 'thresholded_random_geometric_graph({}, {}, {}, {})'
G.name = namestr % (n, radius, theta, dim,)
G.add_nodes_from(nodes)
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = dict((v, seed.expovariate(1)) for v in G)
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = dict((v, [seed.random() for i in range(dim)]) for v in nodes)
# If no distance metric is provided, use Euclidean distance.
nx.set_node_attributes(G, weight, 'weight')
nx.set_node_attributes(G, pos, 'pos')
# Returns ``True`` if and only if the nodes whose attributes are
# ``du`` and ``dv`` should be joined, according to the threshold
# condition and node pairs are within the maximum connection
# distance, ``radius``.
def should_join(pair):
u, v = pair
u_weight, v_weight = weight[u], weight[v]
u_pos, v_pos = pos[u], pos[v]
dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos)))**(1 / p)
# Check if dist is <= radius parameter. This check is redundant if
# scipy is available and _fast_edges routine is used, but provides
# the check in case scipy is not available and all edge combinations
# need to be checked
if dist <= radius:
return theta <= u_weight + v_weight
else:
return False
if _is_scipy_available:
edges = _fast_edges(G, radius, p)
G.add_edges_from(filter(should_join, edges))
else:
G.add_edges_from(filter(should_join, combinations(G, 2)))
return G
def waxman_graph(n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1),
metric=None, seed=None):
r"""Return a Waxman random graph.
The Waxman random graph model places `n` nodes uniformly at random
in a rectangular domain. Each pair of nodes at distance `d` is
joined by an edge with probability
.. math::
p = \beta \exp(-d / \alpha L).
This function implements both Waxman models, using the `L` keyword
argument.
* Waxman-1: if `L` is not specified, it is set to be the maximum distance
between any pair of nodes.
* Waxman-2: if `L` is specified, the distance between a pair of nodes is
chosen uniformly at random from the interval `[0, L]`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
beta: float
Model parameter
alpha: float
Model parameter
L : float, optional
Maximum distance between nodes. If not specified, the actual distance
is calculated.
domain : four-tuple of numbers, optional
Domain size, given as a tuple of the form `(x_min, y_min, x_max,
y_max)`.
metric : function
A metric on vectors of numbers (represented as lists or
tuples). This must be a function that accepts two lists (or
tuples) as input and yields a number as output. The function
must also satisfy the four requirements of a `metric`_.
Specifically, if $d$ is the function and $x$, $y$,
and $z$ are vectors in the graph, then $d$ must satisfy
1. $d(x, y) \ge 0$,
2. $d(x, y) = 0$ if and only if $x = y$,
3. $d(x, y) = d(y, x)$,
4. $d(x, z) \le d(x, y) + d(y, z)$.
If this argument is not specified, the Euclidean distance metric is
used.
.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
Graph
A random Waxman graph, undirected and without self-loops. Each
node has a node attribute ``'pos'`` that stores the position of
that node in Euclidean space as generated by this function.
Examples
--------
Specify an alternate distance metric using the ``metric`` keyword
argument. For example, to use the "`taxicab metric`_" instead of the
default `Euclidean metric`_::
>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
>>> G = nx.waxman_graph(10, 0.5, 0.1, metric=dist)
.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
Notes
-----
Starting in NetworkX 2.0 the parameters alpha and beta align with their
usual roles in the probability distribution. In earlier versions their
positions in the expression were reversed. Their position in the calling
sequence reversed as well to minimize backward incompatibility.
References
----------
.. [1] B. M. Waxman, *Routing of multipoint connections*.
IEEE J. Select. Areas Commun. 6(9),(1988) 1617--1622.
"""
n_name, nodes = n
G = nx.Graph()
G.add_nodes_from(nodes)
(xmin, ymin, xmax, ymax) = domain
# Each node gets a uniformly random position in the given rectangle.
pos = dict((v, (seed.uniform(xmin, xmax), seed.uniform(ymin, ymax))) for v in G)
nx.set_node_attributes(G, pos, 'pos')
# If no distance metric is provided, use Euclidean distance.
if metric is None:
metric = euclidean
# If the maximum distance L is not specified (that is, we are in the
# Waxman-1 model), then find the maximum distance between any pair
# of nodes.
#
# In the Waxman-1 model, join nodes randomly based on distance. In
# the Waxman-2 model, join randomly based on random l.
if L is None:
L = max(metric(x, y) for x, y in combinations(pos.values(), 2))
def dist(u, v): return metric(pos[u], pos[v])
else:
def dist(u, v): return seed.random() * L
# `pair` is the pair of nodes to decide whether to join.
def should_join(pair):
return seed.random() < beta * math.exp(-dist(*pair) / (alpha * L))
G.add_edges_from(filter(should_join, combinations(G, 2)))
return G
def eulerize(G):
"""
Transforms a graph into an Eulerian graph
Parameters
----------
G : NetworkX graph
An undirected graph
Returns
-------
G : NetworkX multigraph
Raises
------
NetworkXError
If the graph is not connected.
See Also
--------
is_eulerian, eulerian_circuit
References
----------
.. [1] J. Edmonds, E. L. Johnson.
Matching, Euler tours and the Chinese postman.
Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
[2] https://en.wikipedia.org/wiki/Eulerian_path
.. [3] http://web.math.princeton.edu/math_alive/5/Notes1.pdf
Examples
--------
>>> G = nx.complete_graph(10)
>>> H = nx.eulerize(G)
>>> nx.is_eulerian(H)
True
"""
if G.order() == 0:
raise nx.NetworkXPointlessConcept("Cannot Eulerize null graph")
if not nx.is_connected(G):
raise nx.NetworkXError("G is not connected")
odd_degree_nodes = [n for n, d in G.degree() if d % 2 == 1]
G = nx.MultiGraph(G)
if len(odd_degree_nodes) == 0:
return G
# get all shortest paths between vertices of odd degree
odd_deg_pairs_paths = [(m, {
n: nx.shortest_path(G, source=m, target=n)
}) for m, n in combinations(odd_degree_nodes, 2)]
# use inverse path lengths as edge-weights in a new graph
# store the paths in the graph for easy indexing later
Gp = nx.Graph()
for n, Ps in odd_deg_pairs_paths:
for m, P in Ps.items():
if n != m:
Gp.add_edge(m, n, weight=1 / len(P), path=P)
# find the minimum weight matching of edges in the weighted graph
best_matching = nx.Graph(list(nx.max_weight_matching(Gp)))
# duplicate each edge along each path in the set of paths in Gp
for m, n in best_matching.edges():
path = Gp[m][n]["path"]
G.add_edges_from(nx.utils.pairwise(path))
return G
def geographical_threshold_graph(n, theta, dim=2, pos=None, weight=None,
metric=None, p_dist=None, seed=None):
r"""Returns a geographical threshold graph.
The geographical threshold graph model places $n$ nodes uniformly at
random in a rectangular domain. Each node $u$ is assigned a weight
$w_u$. Two nodes $u$ and $v$ are joined by an edge if
.. math::
(w_u + w_v)h(r) \ge \theta
where `r` is the distance between `u` and `v`, h(r) is a probability of
connection as a function of `r`, and :math:`\theta` as the threshold
parameter. h(r) corresponds to the p_dist parameter.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict
Node positions as a dictionary of tuples keyed by node.
weight : dict
Node weights as a dictionary of numbers keyed by node.
metric : function
A metric on vectors of numbers (represented as lists or
tuples). This must be a function that accepts two lists (or
tuples) as input and yields a number as output. The function
must also satisfy the four requirements of a `metric`_.
Specifically, if $d$ is the function and $x$, $y$,
and $z$ are vectors in the graph, then $d$ must satisfy
1. $d(x, y) \ge 0$,
2. $d(x, y) = 0$ if and only if $x = y$,
3. $d(x, y) = d(y, x)$,
4. $d(x, z) \le d(x, y) + d(y, z)$.
If this argument is not specified, the Euclidean distance metric is
used.
.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
p_dist : function, optional
A probability density function computing the probability of
connecting two nodes that are of distance, r, computed by metric.
The probability density function, `p_dist`, must
be any function that takes the metric value as input
and outputs a single probability value between 0-1.
The scipy.stats package has many probability distribution functions
implemented and tools for custom probability distribution
definitions [2], and passing the .pdf method of scipy.stats
distributions can be used here. If the probability
function, `p_dist`, is not supplied, the default exponential function
:math: `r^{-2}` is used.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
Graph
A random geographic threshold graph, undirected and without
self-loops.
Each node has a node attribute ``pos`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a node
attribute ``weight`` that stores the weight of that node as
provided or as generated.
Examples
--------
Specify an alternate distance metric using the ``metric`` keyword
argument. For example, to use the `taxicab metric`_ instead of the
default `Euclidean metric`_::
>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
>>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist)
.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
Notes
-----
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter $\lambda=1$.
To specify weights from a different distribution, use the `weight` keyword
argument::
>>> import random
>>> n = 20
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.geographical_threshold_graph(20, 50, weight=w)
If node positions are not specified they are randomly assigned from the
uniform distribution.
Starting in NetworkX 2.1 the parameter ``alpha`` is deprecated and replaced
with the customizable ``p_dist`` function parameter, which defaults to r^-2
if ``p_dist`` is not supplied. To reproduce networks of earlier NetworkX
versions, a custom function needs to be defined and passed as the
``p_dist`` parameter. For example, if the parameter ``alpha`` = 2 was used
in NetworkX 2.0, the custom function def custom_dist(r): r**-2 can be
passed in versions >=2.1 as the parameter p_dist = custom_dist to
produce an equivalent network. Note the change in sign from +2 to -2 in
this parameter change.
References
----------
.. [1] Masuda, N., Miwa, H., Konno, N.:
Geographical threshold graphs with small-world and scale-free
properties.
Physical Review E 71, 036108 (2005)
.. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus,
Giant component and connectivity in geographical threshold graphs,
in Algorithms and Models for the Web-Graph (WAW 2007),
Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007
"""
n_name, nodes = n
G = nx.Graph()
G.add_nodes_from(nodes)
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = dict((v, seed.expovariate(1)) for v in G)
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = dict((v, [seed.random() for i in range(dim)]) for v in nodes)
# If no distance metric is provided, use Euclidean distance.
if metric is None:
metric = euclidean
nx.set_node_attributes(G, weight, 'weight')
nx.set_node_attributes(G, pos, 'pos')
# if p_dist is not supplied, use default r^-2
if p_dist is None:
def p_dist(r):
return r**-2
# Returns ``True`` if and only if the nodes whose attributes are
# ``du`` and ``dv`` should be joined, according to the threshold
# condition.
def should_join(pair):
u, v = pair
u_pos, v_pos = pos[u], pos[v]
u_weight, v_weight = weight[u], weight[v]
return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta
G.add_edges_from(filter(should_join, combinations(G, 2)))
return G
