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 to_prufer_sequence(T):
r"""Returns the Prüfer sequence of the given tree.
A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and
*n* - 1, inclusive. The tree corresponding to a given Prüfer
sequence can be recovered by repeatedly joining a node in the
sequence with a node with the smallest potential degree according to
the sequence.
Parameters
----------
T : NetworkX graph
An undirected graph object representing a tree.
Returns
-------
list
The Prüfer sequence of the given tree.
Raises
------
NetworkXPointlessConcept
If the number of nodes in `T` is less than two.
NotATree
If `T` is not a tree.
KeyError
If the set of nodes in `T` is not {0, …, *n* - 1}.
Notes
-----
There is a bijection from labeled trees to Prüfer sequences. This
function is the inverse of the :func:`from_prufer_sequence`
function.
Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead
of from 0 to *n* - 1. This function requires nodes to be labeled in
the latter form. You can use :func:`~networkx.relabel_nodes` to
relabel the nodes of your tree to the appropriate format.
This implementation is from [1]_ and has a running time of
$O(n \log n)$.
See also
--------
to_nested_tuple
from_prufer_sequence
References
----------
.. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.
"An optimal algorithm for Prufer codes."
*Journal of Software Engineering and Applications* 2.02 (2009): 111.
<http://dx.doi.org/10.4236/jsea.2009.22016>
Examples
--------
There is a bijection between Prüfer sequences and labeled trees, so
this function is the inverse of the :func:`from_prufer_sequence`
function:
>>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
>>> tree = nx.Graph(edges)
>>> sequence = nx.to_prufer_sequence(tree)
>>> sequence
[3, 3, 3, 4]
>>> tree2 = nx.from_prufer_sequence(sequence)
>>> list(tree2.edges()) == edges
True
"""
# Perform some sanity checks on the input.
n = len(T)
if n < 2:
msg = 'Prüfer sequence undefined for trees with fewer than two nodes'
raise nx.NetworkXPointlessConcept(msg)
if not nx.is_tree(T):
raise nx.NotATree('provided graph is not a tree')
if set(T) != set(range(n)):
raise KeyError('tree must have node labels {0, ..., n - 1}')
degree = dict(T.degree())
def parents(u):
return next(v for v in T[u] if degree[v] > 1)
index = u = min(k for k in range(n) if degree[k] == 1)
result = []
for i in range(n - 2):
v = parents(u)
result.append(v)
degree[v] -= 1
if v < index and degree[v] == 1:
u = v
else:
index = u = min(k for k in range(index + 1, n) if degree[k] == 1)
return result
def average_shortest_path_length(G, weight=None, method=None):
r"""Returns the average shortest path length.
The average shortest path length is
.. math::
a =\sum_{s,t \in V} \frac{d(s, t)}{n(n-1)}
where `V` is the set of nodes in `G`,
`d(s, t)` is the shortest path from `s` to `t`,
and `n` is the number of nodes in `G`.
Parameters
----------
G : NetworkX graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
method : string, optional (default = 'unweighted' or 'djikstra')
The algorithm to use to compute the path lengths.
Supported options are 'unweighted', 'dijkstra', 'bellman-ford',
'floyd-warshall' and 'floyd-warshall-numpy'.
Other method values produce a ValueError.
The default method is 'unweighted' if `weight` is None,
otherwise the default method is 'dijkstra'.
Raises
------
NetworkXPointlessConcept
If `G` is the null graph (that is, the graph on zero nodes).
NetworkXError
If `G` is not connected (or not weakly connected, in the case
of a directed graph).
ValueError
If `method` is not among the supported options.
Examples
--------
>>> G = nx.path_graph(5)
>>> nx.average_shortest_path_length(G)
2.0
For disconnected graphs, you can compute the average shortest path
length for each component
>>> G = nx.Graph([(1, 2), (3, 4)])
>>> for C in (G.subgraph(c).copy() for c in nx.connected_components(G)):
... print(nx.average_shortest_path_length(C))
1.0
1.0
"""
single_source_methods = ["unweighted", "dijkstra", "bellman-ford"]
all_pairs_methods = ["floyd-warshall", "floyd-warshall-numpy"]
supported_methods = single_source_methods + all_pairs_methods
if method is None:
method = "unweighted" if weight is None else "dijkstra"
if method not in supported_methods:
raise ValueError(f"method not supported: {method}")
n = len(G)
# For the special case of the null graph, raise an exception, since
# there are no paths in the null graph.
if n == 0:
msg = (
"the null graph has no paths, thus there is no average"
"shortest path length"
)
raise nx.NetworkXPointlessConcept(msg)
# For the special case of the trivial graph, return zero immediately.
if n == 1:
return 0
# Shortest path length is undefined if the graph is disconnected.
if G.is_directed() and not nx.is_weakly_connected(G):
raise nx.NetworkXError("Graph is not weakly connected.")
if not G.is_directed() and not nx.is_connected(G):
raise nx.NetworkXError("Graph is not connected.")
# Compute all-pairs shortest paths.
def path_length(v):
if method == "unweighted":
return nx.single_source_shortest_path_length(G, v)
elif method == "dijkstra":
return nx.single_source_dijkstra_path_length(G, v, weight=weight)
elif method == "bellman-ford":
return nx.single_source_bellman_ford_path_length(G, v, weight=weight)
if method in single_source_methods:
# Sum the distances for each (ordered) pair of source and target node.
s = sum(l for u in G for l in path_length(u).values())
else:
if method == "floyd-warshall":
all_pairs = nx.floyd_warshall(G, weight=weight)
s = sum([sum(t.values()) for t in all_pairs.values()])
elif method == "floyd-warshall-numpy":
s = nx.floyd_warshall_numpy(G, weight=weight).sum()
return s / (n * (n - 1))
def random_tree(n, seed=None, create_using=None):
"""Returns a uniformly random tree on `n` nodes.
Parameters
----------
n : int
A positive integer representing the number of nodes in the tree.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX graph
A tree, given as an undirected graph, whose nodes are numbers in
the set {0, …, *n* - 1}.
Raises
------
NetworkXPointlessConcept
If `n` is zero (because the null graph is not a tree).
Notes
-----
The current implementation of this function generates a uniformly
random Prüfer sequence then converts that to a tree via the
:func:`~networkx.from_prufer_sequence` function. Since there is a
bijection between Prüfer sequences of length *n* - 2 and trees on
*n* nodes, the tree is chosen uniformly at random from the set of
all trees on *n* nodes.
Example
-------
>>> tree = nx.random_tree(n=10, seed=0)
>>> print(nx.forest_str(tree, sources=[0]))
╙── 0
├── 3
└── 4
├── 6
│ ├── 1
│ ├── 2
│ └── 7
│ └── 8
│ └── 5
└── 9
>>> tree = nx.random_tree(n=10, seed=0, create_using=nx.DiGraph)
>>> print(nx.forest_str(tree))
╙── 0
├─╼ 3
└─╼ 4
├─╼ 6
│ ├─╼ 1
│ ├─╼ 2
│ └─╼ 7
│ └─╼ 8
│ └─╼ 5
└─╼ 9
"""
if n == 0:
raise nx.NetworkXPointlessConcept("the null graph is not a tree")
# Cannot create a Prüfer sequence unless `n` is at least two.
if n == 1:
utree = nx.empty_graph(1, create_using)
else:
sequence = [seed.choice(range(n)) for i in range(n - 2)]
utree = nx.from_prufer_sequence(sequence)
if create_using is None:
tree = utree
else:
tree = nx.empty_graph(0, create_using)
if tree.is_directed():
# Use a arbitrary root node and dfs to define edge directions
edges = nx.dfs_edges(utree, source=0)
else:
edges = utree.edges
# Populate the specified graph type
tree.add_nodes_from(utree.nodes)
tree.add_edges_from(edges)
return tree
def average_shortest_path_length(G, weight=None):
r"""Return the average shortest path length.
The average shortest path length is
.. math::
a =\sum_{s,t \in V} \frac{d(s, t)}{n(n-1)}
where `V` is the set of nodes in `G`,
`d(s, t)` is the shortest path from `s` to `t`,
and `n` is the number of nodes in `G`.
Parameters
----------
G : NetworkX graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
Raises
------
NetworkXPointlessConcept
If `G` is the null graph (that is, the graph on zero nodes).
NetworkXError
If `G` is not connected (or not weakly connected, in the case
of a directed graph).
Examples
--------
>>> G = nx.path_graph(5)
>>> nx.average_shortest_path_length(G)
2.0
For disconnected graphs, you can compute the average shortest path
length for each component
>>> G = nx.Graph([(1, 2), (3, 4)])
>>> for C in nx.connected_component_subgraphs(G):
... print(nx.average_shortest_path_length(C))
1.0
1.0
"""
n = len(G)
# For the special case of the null graph, raise an exception, since
# there are no paths in the null graph.
if n == 0:
msg = ('the null graph has no paths, thus there is no average'
'shortest path length')
raise nx.NetworkXPointlessConcept(msg)
# For the special case of the trivial graph, return zero immediately.
if n == 1:
return 0
# Shortest path length is undefined if the graph is disconnected.
if G.is_directed() and not nx.is_weakly_connected(G):
raise nx.NetworkXError("Graph is not weakly connected.")
if not G.is_directed() and not nx.is_connected(G):
raise nx.NetworkXError("Graph is not connected.")
# Compute all-pairs shortest paths.
if weight is None:
def path_length(v):
return nx.single_source_shortest_path_length(G, v)
else:
ssdpl = nx.single_source_dijkstra_path_length
def path_length(v):
return ssdpl(G, v, weight=weight)
# Sum the distances for each (ordered) pair of source and target node.
s = sum(l for u in G for l in path_length(u).values())
return s / (n * (n - 1))
def tree_all_pairs_lowest_common_ancestor(G, root=None, pairs=None):
r"""Yield the lowest common ancestor for sets of pairs in a tree.
Parameters
----------
G : NetworkX directed graph (must be a tree)
root : node, optional (default: None)
The root of the subtree to operate on.
If None, assume the entire graph has exactly one source and use that.
pairs : iterable or iterator of pairs of nodes, optional (default: None)
The pairs of interest. If None, Defaults to all pairs of nodes
under `root` that have a lowest common ancestor.
Returns
-------
lcas : generator of tuples `((u, v), lca)` where `u` and `v` are nodes
in `pairs` and `lca` is their lowest common ancestor.
Notes
-----
Only defined on non-null trees represented with directed edges from
parents to children. Uses Tarjan's off-line lowest-common-ancestors
algorithm. Runs in time $O(4 \times (V + E + P))$ time, where 4 is the largest
value of the inverse Ackermann function likely to ever come up in actual
use, and $P$ is the number of pairs requested (or $V^2$ if all are needed).
Tarjan, R. E. (1979), "Applications of path compression on balanced trees",
Journal of the ACM 26 (4): 690-715, doi:10.1145/322154.322161.
See Also
--------
all_pairs_lowest_common_ancestor (similar routine for general DAGs)
lowest_common_ancestor (just a single pair for general DAGs)
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept("LCA meaningless on null graphs.")
elif None in G:
raise nx.NetworkXError("None is not a valid node.")
# Index pairs of interest for efficient lookup from either side.
if pairs is not None:
pair_dict = defaultdict(set)
# See note on all_pairs_lowest_common_ancestor.
if not isinstance(pairs, (Mapping, Set)):
pairs = set(pairs)
for u, v in pairs:
for n in (u, v):
if n not in G:
msg = "The node %s is not in the digraph." % str(n)
raise nx.NodeNotFound(msg)
pair_dict[u].add(v)
pair_dict[v].add(u)
# If root is not specified, find the exactly one node with in degree 0 and
# use it. Raise an error if none are found, or more than one is. Also check
# for any nodes with in degree larger than 1, which would imply G is not a
# tree.
if root is None:
for n, deg in G.in_degree:
if deg == 0:
if root is not None:
msg = "No root specified and tree has multiple sources."
raise nx.NetworkXError(msg)
root = n
elif deg > 1:
msg = "Tree LCA only defined on trees; use DAG routine."
raise nx.NetworkXError(msg)
if root is None:
raise nx.NetworkXError("Graph contains a cycle.")
# Iterative implementation of Tarjan's offline lca algorithm
# as described in CLRS on page 521.
uf = UnionFind()
ancestors = {}
for node in G:
ancestors[node] = uf[node]
colors = defaultdict(bool)
for node in nx.dfs_postorder_nodes(G, root):
colors[node] = True
for v in (pair_dict[node] if pairs is not None else G):
if colors[v]:
# If the user requested both directions of a pair, give it.
# Otherwise, just give one.
if pairs is not None and (node, v) in pairs:
yield (node, v), ancestors[uf[v]]
if pairs is None or (v, node) in pairs:
yield (v, node), ancestors[uf[v]]
if node != root:
parent = arbitrary_element(G.pred[node])
uf.union(parent, node)
ancestors[uf[parent]] = parent
def all_pairs_lowest_common_ancestor(G, pairs=None):
"""Compute the lowest common ancestor for pairs of nodes.
Parameters
----------
G : NetworkX directed graph
pairs : iterable of pairs of nodes, optional (default: all pairs)
The pairs of nodes of interest.
If None, will find the LCA of all pairs of nodes.
Returns
-------
An iterator over ((node1, node2), lca) where (node1, node2) are
the pairs specified and lca is a lowest common ancestor of the pair.
Note that for the default of all pairs in G, we consider
unordered pairs, e.g. you will not get both (b, a) and (a, b).
Notes
-----
Only defined on non-null directed acyclic graphs.
Uses the $O(n^3)$ ancestor-list algorithm from:
M. A. Bender, M. Farach-Colton, G. Pemmasani, S. Skiena, P. Sumazin.
"Lowest common ancestors in trees and directed acyclic graphs."
Journal of Algorithms, 57(2): 75-94, 2005.
See Also
--------
tree_all_pairs_lowest_common_ancestor
lowest_common_ancestor
"""
if not nx.is_directed_acyclic_graph(G):
raise nx.NetworkXError("LCA only defined on directed acyclic graphs.")
elif len(G) == 0:
raise nx.NetworkXPointlessConcept("LCA meaningless on null graphs.")
elif None in G:
raise nx.NetworkXError("None is not a valid node.")
# The copy isn't ideal, neither is the switch-on-type, but without it users
# passing an iterable will encounter confusing errors, and itertools.tee
# does not appear to handle builtin types efficiently (IE, it materializes
# another buffer rather than just creating listoperators at the same
# offset). The Python documentation notes use of tee is unadvised when one
# is consumed before the other.
#
# This will always produce correct results and avoid unnecessary
# copies in many common cases.
#
if (not isinstance(pairs, (Mapping, Set)) and pairs is not None):
pairs = set(pairs)
# Convert G into a dag with a single root by adding a node with edges to
# all sources iff necessary.
sources = [n for n, deg in G.in_degree if deg == 0]
if len(sources) == 1:
root = sources[0]
super_root = None
else:
G = G.copy()
super_root = root = generate_unique_node()
for source in sources:
G.add_edge(root, source)
# Start by computing a spanning tree, and the DAG of all edges not in it.
# We will then use the tree lca algorithm on the spanning tree, and use
# the DAG to figure out the set of tree queries necessary.
spanning_tree = nx.dfs_tree(G, root)
dag = nx.DiGraph((u, v) for u, v in G.edges
if u not in spanning_tree or v not in spanning_tree[u])
# Ensure that both the dag and the spanning tree contains all nodes in G,
# even nodes that are disconnected in the dag.
spanning_tree.add_nodes_from(G)
dag.add_nodes_from(G)
counter = count()
# Necessary to handle graphs consisting of a single node and no edges.
root_distance = {root: next(counter)}
for edge in nx.bfs_edges(spanning_tree, root):
for node in edge:
if node not in root_distance:
root_distance[node] = next(counter)
# Index the position of all nodes in the Euler tour so we can efficiently
# sort lists and merge in tour order.
euler_tour_pos = {}
for node in nx.depth_first_search.dfs_preorder_nodes(G, root):
if node not in euler_tour_pos:
euler_tour_pos[node] = next(counter)
# Generate the set of all nodes of interest in the pairs.
pairset = set()
if pairs is not None:
pairset = set(chain.from_iterable(pairs))
for n in pairset:
if n not in G:
msg = "The node %s is not in the digraph." % str(n)
raise nx.NodeNotFound(msg)
# Generate the transitive closure over the dag (not G) of all nodes, and
# sort each node's closure set by order of first appearance in the Euler
# tour.
ancestors = {}
for v in dag:
if pairs is None or v in pairset:
my_ancestors = nx.dag.ancestors(dag, v)
my_ancestors.add(v)
ancestors[v] = sorted(my_ancestors, key=euler_tour_pos.get)
def _compute_dag_lca_from_tree_values(tree_lca, dry_run):
"""Iterate through the in-order merge for each pair of interest.
We do this to answer the user's query, but it is also used to
avoid generating unnecessary tree entries when the user only
needs some pairs.
"""
for (node1, node2) in pairs if pairs is not None else tree_lca:
best_root_distance = None
best = None
indices = [0, 0]
ancestors_by_index = [ancestors[node1], ancestors[node2]]
def get_next_in_merged_lists(indices):
"""Returns index of the list containing the next item
Next order refers to the merged order.
Index can be 0 or 1 (or None if exhausted).
"""
index1, index2 = indices
if (index1 >= len(ancestors[node1]) and
index2 >= len(ancestors[node2])):
return None
elif index1 >= len(ancestors[node1]):
return 1
elif index2 >= len(ancestors[node2]):
return 0
elif (euler_tour_pos[ancestors[node1][index1]] <
euler_tour_pos[ancestors[node2][index2]]):
return 0
else:
return 1
# Find the LCA by iterating through the in-order merge of the two
# nodes of interests' ancestor sets. In principle, we need to
# consider all pairs in the Cartesian product of the ancestor sets,
# but by the restricted min range query reduction we are guaranteed
# that one of the pairs of interest is adjacent in the merged list
# iff one came from each list.
i = get_next_in_merged_lists(indices)
cur = ancestors_by_index[i][indices[i]], i
while i is not None:
prev = cur
indices[i] += 1
i = get_next_in_merged_lists(indices)
if i is not None:
cur = ancestors_by_index[i][indices[i]], i
# Two adjacent entries must not be from the same list
# in order for their tree LCA to be considered.
if cur[1] != prev[1]:
tree_node1, tree_node2 = prev[0], cur[0]
if (tree_node1, tree_node2) in tree_lca:
ans = tree_lca[tree_node1, tree_node2]
else:
ans = tree_lca[tree_node2, tree_node1]
if not dry_run and (best is None or
root_distance[ans] > best_root_distance):
best_root_distance = root_distance[ans]
best = ans
# If the LCA is super_root, there is no LCA in the user's graph.
if not dry_run and (super_root is None or best != super_root):
yield (node1, node2), best
# Generate the spanning tree lca for all pairs. This doesn't make sense to
# do incrementally since we are using a linear time offline algorithm for
# tree lca.
if pairs is None:
# We want all pairs so we'll need the entire tree.
tree_lca = dict(tree_all_pairs_lowest_common_ancestor(spanning_tree,
root))
else:
# We only need the merged adjacent pairs by seeing which queries the
# algorithm needs then generating them in a single pass.
tree_lca = defaultdict(int)
for _ in _compute_dag_lca_from_tree_values(tree_lca, True):
pass
# Replace the bogus default tree values with the real ones.
for (pair, lca) in tree_all_pairs_lowest_common_ancestor(spanning_tree,
root,
tree_lca):
tree_lca[pair] = lca
# All precomputations complete. Now we just need to give the user the pairs
# they asked for, or all pairs if they want them all.
return _compute_dag_lca_from_tree_values(tree_lca, False)
def eigenvector_centrality(G,
max_iter=100,
tol=1.0e-6,
nstart=None,
weight=None):
r"""Compute the eigenvector centrality for the graph `G`.
Eigenvector centrality computes the centrality for a node based on the
centrality of its neighbors. The eigenvector centrality for node $i$ is
the $i$-th element of the vector $x$ defined by the equation
.. math::
Ax = \lambda x
where $A$ is the adjacency matrix of the graph `G` with eigenvalue
$\lambda$. By virtue of the Perron–Frobenius theorem, there is a unique
solution $x$, all of whose entries are positive, if $\lambda$ is the
largest eigenvalue of the adjacency matrix $A$ ([2]_).
Parameters
----------
G : graph
A networkx graph
max_iter : integer, optional (default=100)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional (default=None)
Starting value of eigenvector iteration for each node.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Returns
-------
nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
Examples
--------
>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality(G)
>>> sorted((v, '{:0.2f}'.format(c)) for v, c in centrality.items())
[(0, '0.37'), (1, '0.60'), (2, '0.60'), (3, '0.37')]
Raises
------
NetworkXPointlessConcept
If the graph `G` is the null graph.
NetworkXError
If each value in `nstart` is zero.
PowerIterationFailedConvergence
If the algorithm fails to converge to the specified tolerance
within the specified number of iterations of the power iteration
method.
See Also
--------
eigenvector_centrality_numpy
pagerank
hits
Notes
-----
The measure was introduced by [1]_ and is discussed in [2]_.
The power iteration method is used to compute the eigenvector and
convergence is **not** guaranteed. Our method stops after ``max_iter``
iterations or when the change in the computed vector between two
iterations is smaller than an error tolerance of
``G.number_of_nodes() * tol``. This implementation uses ($A + I$)
rather than the adjacency matrix $A$ because it shifts the spectrum
to enable discerning the correct eigenvector even for networks with
multiple dominant eigenvalues.
For directed graphs this is "left" eigenvector centrality which corresponds
to the in-edges in the graph. For out-edges eigenvector centrality
first reverse the graph with ``G.reverse()``.
References
----------
.. [1] Phillip Bonacich.
"Power and Centrality: A Family of Measures."
*American Journal of Sociology* 92(5):1170–1182, 1986
<http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf>
.. [2] Mark E. J. Newman.
*Networks: An Introduction.*
Oxford University Press, USA, 2010, pp. 169.
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept('cannot compute centrality for the'
' null graph')
# If no initial vector is provided, start with the all-ones vector.
if nstart is None:
nstart = {v: 1 for v in G}
if all(v == 0 for v in nstart.values()):
raise nx.NetworkXError('initial vector cannot have all zero values')
# Normalize the initial vector so that each entry is in [0, 1]. This is
# guaranteed to never have a divide-by-zero error by the previous line.
nstart_sum = sum(nstart.values())
x = {k: v / nstart_sum for k, v in nstart.items()}
nnodes = G.number_of_nodes()
# make up to max_iter iterations
for i in range(max_iter):
xlast = x
x = xlast.copy() # Start with xlast times I to iterate with (A+I)
# do the multiplication y^T = x^T A (left eigenvector)
for n in x:
for nbr in G[n]:
w = G[n][nbr].get(weight, 1) if weight else 1
x[nbr] += xlast[n] * w
# Normalize the vector. The normalization denominator `norm`
# should never be zero by the Perron--Frobenius
# theorem. However, in case it is due to numerical error, we
# assume the norm to be one instead.
norm = sqrt(sum(z**2 for z in x.values())) or 1
x = {k: v / norm for k, v in x.items()}
# Check for convergence (in the L_1 norm).
if sum(abs(x[n] - xlast[n]) for n in x) < nnodes * tol:
return x
raise nx.PowerIterationFailedConvergence(max_iter)
def eigenvector_centrality_numpy(G, weight=None, max_iter=50, tol=0):
r"""Compute the eigenvector centrality for the graph G.
Eigenvector centrality computes the centrality for a node based on the
centrality of its neighbors. The eigenvector centrality for node $i$ is
.. math::
Ax = \lambda x
where $A$ is the adjacency matrix of the graph G with eigenvalue $\lambda$.
By virtue of the Perron–Frobenius theorem, there is a unique and positive
solution if $\lambda$ is the largest eigenvalue associated with the
eigenvector of the adjacency matrix $A$ ([2]_).
Parameters
----------
G : graph
A networkx graph
weight : None or string, optional (default=None)
The name of the edge attribute used as weight.
If None, all edge weights are considered equal.
max_iter : integer, optional (default=100)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Relative accuracy for eigenvalues (stopping criterion).
The default value of 0 implies machine precision.
Returns
-------
nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
Examples
--------
>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality_numpy(G)
>>> print(['{} {:0.2f}'.format(node, centrality[node]) for node in centrality])
['0 0.37', '1 0.60', '2 0.60', '3 0.37']
See Also
--------
eigenvector_centrality
pagerank
hits
Notes
-----
The measure was introduced by [1]_.
This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to
find the largest eigenvalue/eigenvector pair.
For directed graphs this is "left" eigenvector centrality which corresponds
to the in-edges in the graph. For out-edges eigenvector centrality
first reverse the graph with ``G.reverse()``.
Raises
------
NetworkXPointlessConcept
If the graph ``G`` is the null graph.
References
----------
.. [1] Phillip Bonacich:
Power and Centrality: A Family of Measures.
American Journal of Sociology 92(5):1170–1182, 1986
http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf
.. [2] Mark E. J. Newman:
Networks: An Introduction.
Oxford University Press, USA, 2010, pp. 169.
"""
import scipy as sp
from scipy.sparse import linalg
if len(G) == 0:
raise nx.NetworkXPointlessConcept('cannot compute centrality for the'
' null graph')
M = nx.to_scipy_sparse_matrix(G,
nodelist=list(G),
weight=weight,
dtype=float)
eigenvalue, eigenvector = linalg.eigs(M.T,
k=1,
which='LR',
maxiter=max_iter,
tol=tol)
largest = eigenvector.flatten().real
norm = sp.sign(largest.sum()) * sp.linalg.norm(largest)
return dict(zip(G, largest / norm))
def greedy_color(G, strategy='largest_first', interchange=False):
"""Color a graph using various strategies of greedy graph coloring.
Attempts to color a graph using as few colors as possible, where no
neighbours of a node can have same color as the node itself. The
given strategy determines the order in which nodes are colored.
The strategies are described in [1]_.
Parameters
----------
G : NetworkX graph
strategy : string or function(G, colors)
A function (or a string representing a function) that provides
the coloring strategy, by returning nodes in the ordering they
should be colored. ``G`` is the graph, and ``colors`` is a
dictionary of the currently assigned colors, keyed by nodes. The
function must return an iterable over all the nodes in ``G``.
If the strategy function is an iterator generator (that is, a
function with ``yield`` statements), keep in mind that the
``colors`` dictionary will be updated after each ``yield``, since
this function chooses colors greedily.
If ``strategy`` is a string, it must be one of the following,
each of which represents one of the built-in strategy functions.
* ``'largest_first'``
* ``'random_sequential'``
* ``'smallest_last'``
* ``'independent_set'``
* ``'connected_sequential_bfs'``
* ``'connected_sequential_dfs'``
* ``'connected_sequential'`` (alias for the previous strategy)
* ``'strategy_saturation_largest_first'``
* ``'DSATUR'`` (alias for the previous strategy)
interchange: bool
Will use the color interchange algorithm described by [2]_ if set
to ``True``.
Note that ``strategy_saturation_largest_first`` and
``strategy_independent_set`` do not work with
interchange. Furthermore, if you use interchange with your own
strategy function, you cannot rely on the values in the
``colors`` argument.
Returns
-------
A dictionary with keys representing nodes and values representing
corresponding coloring.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> d = nx.coloring.greedy_color(G, strategy='largest_first')
>>> d in [{0: 0, 1: 1, 2: 0, 3: 1}, {0: 1, 1: 0, 2: 1, 3: 0}]
True
Raises
------
NetworkXPointlessConcept
If ``strategy`` is ``strategy_saturation_largest_first`` or
``strategy_independent_set`` and ``interchange`` is ``True``.
References
----------
.. [1] Adrian Kosowski, and Krzysztof Manuszewski,
Classical Coloring of Graphs, Graph Colorings, 2-19, 2004.
ISBN 0-8218-3458-4.
.. [2] Maciej M. Sysło, Marsingh Deo, Janusz S. Kowalik,
Discrete Optimization Algorithms with Pascal Programs, 415-424, 1983.
ISBN 0-486-45353-7.
"""
if len(G) == 0:
return {}
# Determine the strategy provided by the caller.
strategy = STRATEGIES.get(strategy, strategy)
if not callable(strategy):
raise nx.NetworkXError('strategy must be callable or a valid string. '
'{0} not valid.'.format(strategy))
# Perform some validation on the arguments before executing any
# strategy functions.
if interchange:
if strategy is strategy_independent_set:
msg = 'interchange cannot be used with strategy_independent_set'
raise nx.NetworkXPointlessConcept(msg)
if strategy is strategy_saturation_largest_first:
msg = ('interchange cannot be used with'
' strategy_saturation_largest_first')
raise nx.NetworkXPointlessConcept(msg)
colors = {}
nodes = strategy(G, colors)
if interchange:
return _interchange.greedy_coloring_with_interchange(G, nodes)
for u in nodes:
# Set to keep track of colors of neighbours
neighbour_colors = {colors[v] for v in G[u] if v in colors}
# Find the first unused color.
for color in itertools.count():
if color not in neighbour_colors:
break
# Assign the new color to the current node.
colors[u] = color
return colors