def rooted_product(G, H, root):
""" Return the rooted product of graphs G and H rooted at root in H.
A new graph is constructed representing the rooted product of
the inputted graphs, G and H, with a root in H.
A rooted product duplicates H for each nodes in G with the root
of H corresponding to the node in G. Nodes are renamed as the direct
product of G and H. The result is a subgraph of the cartesian product.
Parameters
----------
G,H : graph
A NetworkX graph
root : node
A node in H
Returns
-------
R : The rooted product of G and H with a specified root in H
Notes
-----
The nodes of R are the Cartesian Product of the nodes of G and H.
The nodes of G and H are not relabeled.
"""
if root not in H:
raise nx.NetworkXError('root must be a vertex in H')
R = nx.Graph()
R.add_nodes_from(product(G, H))
R.add_edges_from(((e[0], root), (e[1], root)) for e in G.edges())
R.add_edges_from(((g, e[0]), (g, e[1])) for g in G for e in H.edges())
return R
def test_inverse(self):
"""Tests that the encoding and decoding functions are inverses.
"""
for T in nx.nonisomorphic_trees(4):
T2 = nx.from_prufer_sequence(nx.to_prufer_sequence(T))
assert_nodes_equal(list(T), list(T2))
assert_edges_equal(list(T.edges()), list(T2.edges()))
for seq in product(range(4), repeat=2):
seq2 = nx.to_prufer_sequence(nx.from_prufer_sequence(seq))
assert_equal(list(seq), seq2)
def root_to_leaf_paths(G):
"""Yields root-to-leaf paths in a directed acyclic graph.
`G` must be a directed acyclic graph. If not, the behavior of this
function is undefined. A "root" in this graph is a node of in-degree
zero and a "leaf" a node of out-degree zero.
When invoked, this function iterates over each path from any root to
any leaf. A path is a list of nodes.
"""
roots = (v for v, d in G.in_degree() if d == 0)
leaves = (v for v, d in G.out_degree() if d == 0)
all_paths = partial(nx.all_simple_paths, G)
# TODO In Python 3, this would be better as `yield from ...`.
return chaini(starmap(all_paths, product(roots, leaves)))
def test_all_pairs_lowest_common_ancestor3(self):
"""Produces the correct results when all pairs given as a generator."""
all_pairs = product(self.DG.nodes(), self.DG.nodes())
ans = all_pairs_lca(self.DG, pairs=all_pairs)
self.assert_lca_dicts_same(dict(ans), self.gold)
def _node_product(G, H):
for u, v in product(G, H):
yield ((u, v), _dict_product(G.nodes[u], H.nodes[v]))
def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None):
r"""Return a navigable small-world graph.
A navigable small-world graph is a directed grid with additional long-range
connections that are chosen randomly.
[...] we begin with a set of nodes [...] that are identified with the set
of lattice points in an $n \times n$ square,
$\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$,
and we define the *lattice distance* between two nodes $(i, j)$ and
$(k, l)$ to be the number of "lattice steps" separating them:
$d((i, j), (k, l)) = |k - i| + |l - j|$.
For a universal constant $p >= 1$, the node $u$ has a directed edge to
every other node within lattice distance $p$---these are its *local
contacts*. For universal constants $q >= 0$ and $r >= 0$ we also
construct directed edges from $u$ to $q$ other nodes (the *long-range
contacts*) using independent random trials; the $i$th directed edge from
$u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$.
-- [1]_
Parameters
----------
n : int
The length of one side of the lattice; the number of nodes in
the graph is therefore $n^2$.
p : int
The diameter of short range connections. Each node is joined with every
other node within this lattice distance.
q : int
The number of long-range connections for each node.
r : float
Exponent for decaying probability of connections. The probability of
connecting to a node at lattice distance $d$ is $1/d^r$.
dim : int
Dimension of grid
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
References
----------
.. [1] J. Kleinberg. The small-world phenomenon: An algorithmic
perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.
"""
if (p < 1):
raise nx.NetworkXException("p must be >= 1")
if (q < 0):
raise nx.NetworkXException("q must be >= 0")
if (r < 0):
raise nx.NetworkXException("r must be >= 1")
G = nx.DiGraph()
nodes = list(product(range(n), repeat=dim))
for p1 in nodes:
probs = [0]
for p2 in nodes:
if p1 == p2:
continue
d = sum((abs(b - a) for a, b in zip(p1, p2)))
if d <= p:
G.add_edge(p1, p2)
probs.append(d**-r)
cdf = list(nx.utils.accumulate(probs))
for _ in range(q):
target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))]
G.add_edge(p1, target)
return G
def edge_relation(b, c):
return any(v in G[u] for u, v in product(b, c))
def _quotient_graph(G,
partition,
edge_relation=None,
node_data=None,
edge_data=None,
relabel=False,
create_using=None):
# Each node in the graph must be in exactly one block.
if any(sum(1 for b in partition if v in b) != 1 for v in G):
raise NetworkXException('each node must be in exactly one block')
if create_using is None:
H = G.__class__()
else:
H = nx.empty_graph(0, create_using)
# By default set some basic information about the subgraph that each block
# represents on the nodes in the quotient graph.
if node_data is None:
def node_data(b):
S = G.subgraph(b)
return dict(graph=S,
nnodes=len(S),
nedges=S.number_of_edges(),
density=density(S))
# Each block of the partition becomes a node in the quotient graph.
partition = [frozenset(b) for b in partition]
H.add_nodes_from((b, node_data(b)) for b in partition)
# By default, the edge relation is the relation defined as follows. B is
# adjacent to C if a node in B is adjacent to a node in C, according to the
# edge set of G.
#
# This is not a particularly efficient implementation of this relation:
# there are O(n^2) pairs to check and each check may require O(log n) time
# (to check set membership). This can certainly be parallelized.
if edge_relation is None:
def edge_relation(b, c):
return any(v in G[u] for u, v in product(b, c))
# By default, sum the weights of the edges joining pairs of nodes across
# blocks to get the weight of the edge joining those two blocks.
if edge_data is None:
def edge_data(b, c):
edgedata = (d for u, v, d in G.edges(b | c, data=True)
if (u in b and v in c) or (u in c and v in b))
return dict(('weight', sum(d.get('weight', 1))) for d in edgedata)
block_pairs = permutations(H, 2) if H.is_directed() else combinations(H, 2)
# In a multigraph, add one edge in the quotient graph for each edge
# in the original graph.
if H.is_multigraph():
edges = chaini(((b, c, G.get_edge_data(u, v, default={}))
for u, v in product(b, c) if v in G[u])
for b, c in block_pairs if edge_relation(b, c))
# In a simple graph, apply the edge data function to each pair of
# blocks to determine the edge data attributes to apply to each edge
# in the quotient graph.
else:
edges = ((b, c, edge_data(b, c)) for (b, c) in block_pairs
if edge_relation(b, c))
H.add_edges_from(edges)
# If requested by the user, relabel the nodes to be integers,
# numbered in increasing order from zero in the same order as the
# iteration order of `partition`.
if relabel:
# Can't use nx.convert_node_labels_to_integers() here since we
# want the order of iteration to be the same for backward
# compatibility with the nx.blockmodel() function.
labels = dict((b, i) for i, b in enumerate(partition))
H = nx.relabel_nodes(H, labels)
return H
def modularity(G, communities, weight='weight'):
r"""Returns the modularity of the given partition of the graph.
Modularity is defined in [1]_ as
.. math::
Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \frac{k_ik_j}{2m}\right)
\delta(c_i,c_j)
where $m$ is the number of edges, $A$ is the adjacency matrix of
`G`, $k_i$ is the degree of $i$ and $\delta(c_i, c_j)$
is 1 if $i$ and $j$ are in the same community and 0 otherwise.
Parameters
----------
G : NetworkX Graph
communities : list
List of sets of nodes of `G` representing a partition of the
nodes.
Returns
-------
Q : float
The modularity of the paritition.
Raises
------
NotAPartition
If `communities` is not a partition of the nodes of `G`.
Examples
--------
>>> G = nx.barbell_graph(3, 0)
>>> nx.algorithms.community.modularity(G, [{0, 1, 2}, {3, 4, 5}])
0.35714285714285704
References
----------
.. [1] M. E. J. Newman *Networks: An Introduction*, page 224.
Oxford University Press, 2011.
"""
if not is_partition(G, communities):
raise NotAPartition(G, communities)
multigraph = G.is_multigraph()
directed = G.is_directed()
m = G.size(weight=weight)
if directed:
out_degree = dict(G.out_degree(weight=weight))
in_degree = dict(G.in_degree(weight=weight))
norm = 1 / m
else:
out_degree = dict(G.degree(weight=weight))
in_degree = out_degree
norm = 1 / (2 * m)
def val(u, v):
try:
if multigraph:
w = sum(d.get(weight, 1) for k, d in G[u][v].items())
else:
w = G[u][v].get(weight, 1)
except KeyError:
w = 0
# Double count self-loops if the graph is undirected.
if u == v and not directed:
w *= 2
return w - in_degree[u] * out_degree[v] * norm
Q = sum(val(u, v) for c in communities for u, v in product(c, repeat=2))
return Q * norm
