# Copyright (C) 2004-2016 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
# Authors: Aric Hagberg <aric.hagberg@gmail.com>,
# Dan Schult <dschult@colgate.edu>
"""
******
Layout
******
Node positioning algorithms for graph drawing.
For `random_layout()` the possible resulting shape
is a square of side [0, scale] (default: [0, 1])
Changing `center` shifts the layout by that amount.
For the other layout routines, the extent is
[center - scale, center + scale] (default: [-1, 1]).
Warning: Most layout routines have only been tested in 2-dimensions.
"""
from __future__ import division
import collections
import networkx as nx
__all__ = ['circular_layout',
'random_layout',
'rescale_layout',
'shell_layout',
'spring_layout',
'spectral_layout',
'fruchterman_reingold_layout']
def _process_params(G, center, dim):
# Some boilerplate code.
import numpy as np
if not isinstance(G, nx.Graph):
empty_graph = nx.Graph()
empty_graph.add_nodes_from(G)
G = empty_graph
if center is None:
center = np.zeros(dim)
else:
center = np.asarray(center)
if len(center) != dim:
msg = "length of center coordinates must match dimension of layout"
raise ValueError(msg)
return G, center
[docs]def random_layout(G, center=None, dim=2):
"""Position nodes uniformly at random in the unit square.
For every node, a position is generated by choosing each of dim
coordinates uniformly at random on the interval [0.0, 1.0).
NumPy (http://scipy.org) is required for this function.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> pos = nx.random_layout(G)
"""
import numpy as np
G, center = _process_params(G, center, dim)
shape = (len(G), dim)
pos = np.random.random(shape) + center
pos = pos.astype(np.float32)
pos = dict(zip(G, pos))
return pos
[docs]def circular_layout(G, scale=1, center=None, dim=2):
# dim=2 only
"""Position nodes on a circle.
Parameters
----------
G : NetworkX graph or list of nodes
scale : float
Scale factor for positions
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout, currently only dim=2 is supported
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.circular_layout(G)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
"""
import numpy as np
G, center = _process_params(G, center, dim)
if len(G) == 0:
pos = {}
elif len(G) == 1:
pos = {nx.utils.arbitrary_element(G): center}
else:
# Discard the extra angle since it matches 0 radians.
theta = np.linspace(0, 1, len(G) + 1)[:-1] * 2 * np.pi
theta = theta.astype(np.float32)
pos = np.column_stack([np.cos(theta), np.sin(theta)])
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos
[docs]def shell_layout(G, nlist=None, scale=1, center=None, dim=2):
"""Position nodes in concentric circles.
Parameters
----------
G : NetworkX graph or list of nodes
nlist : list of lists
List of node lists for each shell.
scale : float
Scale factor for positions
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout, currently only dim=2 is supported
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> shells = [[0], [1, 2, 3]]
>>> pos = nx.shell_layout(G, shells)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
"""
import numpy as np
G, center = _process_params(G, center, dim)
if len(G) == 0:
return {}
if len(G) == 1:
return {nx.utils.arbitrary_element(G): center}
if nlist is None:
# draw the whole graph in one shell
nlist = [list(G)]
if len(nlist[0]) == 1:
# single node at center
radius = 0.0
else:
# else start at r=1
radius = 1.0
npos = {}
for nodes in nlist:
# Discard the extra angle since it matches 0 radians.
theta = np.linspace(0, 1, len(nodes) + 1)[:-1] * 2 * np.pi
theta = theta.astype(np.float32)
pos = np.column_stack([np.cos(theta), np.sin(theta)])
pos = rescale_layout(pos, scale=scale * radius / len(nlist)) + center
npos.update(zip(nodes, pos))
radius += 1.0
return npos
def fruchterman_reingold_layout(G, k=None,
pos=None,
fixed=None,
iterations=50,
weight='weight',
scale=1.0,
center=None,
dim=2):
"""Position nodes using Fruchterman-Reingold force-directed algorithm.
Parameters
----------
G : NetworkX graph or list of nodes
k : float (default=None)
Optimal distance between nodes. If None the distance is set to
1/sqrt(n) where n is the number of nodes. Increase this value
to move nodes farther apart.
pos : dict or None optional (default=None)
Initial positions for nodes as a dictionary with node as keys
and values as a coordinate list or tuple. If None, then use
random initial positions.
fixed : list or None optional (default=None)
Nodes to keep fixed at initial position.
iterations : int optional (default=50)
Number of iterations of spring-force relaxation
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. If None, then all edge weights are 1.
scale : float (default=1.0)
Scale factor for positions. The nodes are positioned
in a box of size [0, scale] x [0, scale].
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.spring_layout(G)
# The same using longer but equivalent function name
>>> pos = nx.fruchterman_reingold_layout(G)
"""
import numpy as np
G, center = _process_params(G, center, dim)
if fixed is not None:
nfixed = dict(zip(G, range(len(G))))
fixed = np.asarray([nfixed[v] for v in fixed])
if pos is not None:
# Determine size of existing domain to adjust initial positions
dom_size = max(coord for pos_tup in pos.values() for coord in pos_tup)
shape = (len(G), dim)
pos_arr = np.random.random(shape) * dom_size + center
for i, n in enumerate(G):
if n in pos:
pos_arr[i] = np.asarray(pos[n])
else:
pos_arr = None
if len(G) == 0:
return {}
if len(G) == 1:
return {next(G.nodes()): center}
try:
# Sparse matrix
if len(G) < 500: # sparse solver for large graphs
raise ValueError
A = nx.to_scipy_sparse_matrix(G, weight=weight, dtype='f')
if k is None and fixed is not None:
# We must adjust k by domain size for layouts not near 1x1
nnodes, _ = A.shape
k = dom_size / np.sqrt(nnodes)
pos = _sparse_fruchterman_reingold(A, k, pos_arr, fixed,
iterations, dim)
except:
A = nx.to_numpy_matrix(G, weight=weight)
if k is None and fixed is not None:
# We must adjust k by domain size for layouts not near 1x1
nnodes, _ = A.shape
k = dom_size / np.sqrt(nnodes)
pos = _fruchterman_reingold(A, k, pos_arr, fixed, iterations, dim)
if fixed is None:
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos
spring_layout = fruchterman_reingold_layout
def _fruchterman_reingold(A, k=None, pos=None, fixed=None,
iterations=50, dim=2):
# Position nodes in adjacency matrix A using Fruchterman-Reingold
# Entry point for NetworkX graph is fruchterman_reingold_layout()
try:
import numpy as np
except ImportError:
msg = "_fruchterman_reingold() requires numpy: http://scipy.org/ "
raise ImportError(msg)
try:
nnodes, _ = A.shape
except AttributeError:
msg = "fruchterman_reingold() takes an adjacency matrix as input"
raise nx.NetworkXError(msg)
# make sure we have an array instead of a matrix
A = np.asarray(A)
if pos is None:
# random initial positions
pos = np.asarray(np.random.random((nnodes, dim)), dtype=A.dtype)
else:
# make sure positions are of same type as matrix
pos = pos.astype(A.dtype)
# optimal distance between nodes
if k is None:
k = np.sqrt(1.0/nnodes)
# the initial "temperature" is about .1 of domain area (=1x1)
# this is the largest step allowed in the dynamics.
# We need to calculate this in case our fixed positions force our domain
# to be much bigger than 1x1
t = max(max(pos.T[0]) - min(pos.T[0]), max(pos.T[1]) - min(pos.T[1]))*0.1
# simple cooling scheme.
# linearly step down by dt on each iteration so last iteration is size dt.
dt = t/float(iterations+1)
delta = np.zeros((pos.shape[0], pos.shape[0], pos.shape[1]), dtype=A.dtype)
# the inscrutable (but fast) version
# this is still O(V^2)
# could use multilevel methods to speed this up significantly
for iteration in range(iterations):
# matrix of difference between points
for i in range(pos.shape[1]):
delta[:, :, i] = pos[:, i, None] - pos[:, i]
# distance between points
distance = np.sqrt((delta**2).sum(axis=-1))
# enforce minimum distance of 0.01
distance = np.where(distance < 0.01, 0.01, distance)
# displacement "force"
displacement = np.transpose(np.transpose(delta) *
(k * k / distance**2 - A * distance / k)
).sum(axis=1)
# update positions
length = np.sqrt((displacement**2).sum(axis=1))
length = np.where(length < 0.01, 0.1, length)
delta_pos = np.transpose(np.transpose(displacement) * t / length)
if fixed is not None:
# don't change positions of fixed nodes
delta_pos[fixed] = 0.0
pos += delta_pos
# cool temperature
t -= dt
return pos
def _sparse_fruchterman_reingold(A, k=None, pos=None, fixed=None,
iterations=50, dim=2):
# Position nodes in adjacency matrix A using Fruchterman-Reingold
# Entry point for NetworkX graph is fruchterman_reingold_layout()
# Sparse version
try:
import numpy as np
except ImportError:
m = "_sparse_fruchterman_reingold() requires numpy: http://scipy.org/"
raise ImportError(m)
try:
nnodes, _ = A.shape
except AttributeError:
msg = "fruchterman_reingold() takes an adjacency matrix as input"
raise nx.NetworkXError(msg)
try:
from scipy.sparse import spdiags, coo_matrix
except ImportError:
msg = "_sparse_fruchterman_reingold() scipy numpy: http://scipy.org/ "
raise ImportError(msg)
# make sure we have a LIst of Lists representation
try:
A = A.tolil()
except:
A = (coo_matrix(A)).tolil()
if pos is None:
# random initial positions
pos = np.asarray(np.random.random((nnodes, dim)), dtype=A.dtype)
else:
# make sure positions are of same type as matrix
pos = pos.astype(A.dtype)
# no fixed nodes
if fixed is None:
fixed = []
# optimal distance between nodes
if k is None:
k = np.sqrt(1.0/nnodes)
# the initial "temperature" is about .1 of domain area (=1x1)
# this is the largest step allowed in the dynamics.
t = 0.1
# simple cooling scheme.
# linearly step down by dt on each iteration so last iteration is size dt.
dt = t / float(iterations+1)
displacement = np.zeros((dim, nnodes))
for iteration in range(iterations):
displacement *= 0
# loop over rows
for i in range(A.shape[0]):
if i in fixed:
continue
# difference between this row's node position and all others
delta = (pos[i] - pos).T
# distance between points
distance = np.sqrt((delta**2).sum(axis=0))
# enforce minimum distance of 0.01
distance = np.where(distance < 0.01, 0.01, distance)
# the adjacency matrix row
Ai = np.asarray(A.getrowview(i).toarray())
# displacement "force"
displacement[:, i] +=\
(delta * (k * k / distance**2 - Ai * distance / k)).sum(axis=1)
# update positions
length = np.sqrt((displacement**2).sum(axis=0))
length = np.where(length < 0.01, 0.1, length)
pos += (displacement * t / length).T
# cool temperature
t -= dt
return pos
[docs]def spectral_layout(G, weight='weight', scale=1, center=None, dim=2):
"""Position nodes using the eigenvectors of the graph Laplacian.
Parameters
----------
G : NetworkX graph or list of nodes
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. If None, then all edge weights are 1.
scale : float
Scale factor for positions
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.spectral_layout(G)
Notes
-----
Directed graphs will be considered as undirected graphs when
positioning the nodes.
For larger graphs (>500 nodes) this will use the SciPy sparse
eigenvalue solver (ARPACK).
"""
# handle some special cases that break the eigensolvers
import numpy as np
G, center = _process_params(G, center, dim)
if len(G) <= 2:
if len(G) == 0:
pos = np.array([])
elif len(G) == 1:
pos = np.array([center])
else:
pos = np.array([np.zeros(dim), np.array(center)*2.0])
return dict(zip(G, pos))
try:
# Sparse matrix
if len(G) < 500: # dense solver is faster for small graphs
raise ValueError
A = nx.to_scipy_sparse_matrix(G, weight=weight, dtype='d')
# Symmetrize directed graphs
if G.is_directed():
A = A + np.transpose(A)
pos = _sparse_spectral(A, dim)
except (ImportError, ValueError):
# Dense matrix
A = nx.to_numpy_matrix(G, weight=weight)
# Symmetrize directed graphs
if G.is_directed():
A = A + np.transpose(A)
pos = _spectral(A, dim)
pos = rescale_layout(pos, scale) + center
pos = dict(zip(G, pos))
return pos
def _spectral(A, dim=2):
# Input adjacency matrix A
# Uses dense eigenvalue solver from numpy
try:
import numpy as np
except ImportError:
msg = "spectral_layout() requires numpy: http://scipy.org/ "
raise ImportError(msg)
try:
nnodes, _ = A.shape
except AttributeError:
msg = "spectral() takes an adjacency matrix as input"
raise nx.NetworkXError(msg)
# form Laplacian matrix
# make sure we have an array instead of a matrix
A = np.asarray(A)
I = np.identity(nnodes, dtype=A.dtype)
D = I * np.sum(A, axis=1) # diagonal of degrees
L = D - A
eigenvalues, eigenvectors = np.linalg.eig(L)
# sort and keep smallest nonzero
index = np.argsort(eigenvalues)[1:dim + 1] # 0 index is zero eigenvalue
return np.real(eigenvectors[:, index])
def _sparse_spectral(A, dim=2):
# Input adjacency matrix A
# Uses sparse eigenvalue solver from scipy
# Could use multilevel methods here, see Koren "On spectral graph drawing"
try:
import numpy as np
from scipy.sparse import spdiags
except ImportError:
msg = "_sparse_spectral() requires scipy & numpy: http://scipy.org/ "
raise ImportError(msg)
try:
from scipy.sparse.linalg.eigen import eigsh
except ImportError:
# scipy <0.9.0 names eigsh differently
from scipy.sparse.linalg import eigen_symmetric as eigsh
try:
nnodes, _ = A.shape
except AttributeError:
msg = "sparse_spectral() takes an adjacency matrix as input"
raise nx.NetworkXError(msg)
# form Laplacian matrix
data = np.asarray(A.sum(axis=1).T)
D = spdiags(data, 0, nnodes, nnodes)
L = D - A
k = dim + 1
# number of Lanczos vectors for ARPACK solver.What is the right scaling?
ncv = max(2 * k + 1, int(np.sqrt(nnodes)))
# return smallest k eigenvalues and eigenvectors
eigenvalues, eigenvectors = eigsh(L, k, which='SM', ncv=ncv)
index = np.argsort(eigenvalues)[1:k] # 0 index is zero eigenvalue
return np.real(eigenvectors[:, index])
[docs]def rescale_layout(pos, scale=1):
"""Return scaled position array to (-scale, scale) in all axes.
The function acts on NumPy arrays which hold position information.
Each position is one row of the array. The dimension of the space
equals the number of columns. Each coordinate in one column.
To rescale, the mean (center) is subtracted from each axis separately.
Then all values are scaled so that the largest magnitude value
from all axes equals `scale` (thus, the aspect ratio is preserved).
The resulting NumPy Array is returned (order of rows unchanged).
Parameters
----------
pos : numpy array
positions to be scaled. Each row is a position.
scale : number (default: 1)
The size of the resulting extent in all directions.
Returns
-------
pos : numpy array
scaled positions. Each row is a position.
"""
# Find max length over all dimensions
lim = 0 # max coordinate for all axes
for i in range(pos.shape[1]):
pos[:, i] -= pos[:, i].mean()
lim = max(pos[:, i].max(), lim)
# rescale to (-scale, scale) in all directions, preserves aspect
if lim > 0:
for i in range(pos.shape[1]):
pos[:, i] *= scale / lim
return pos
# fixture for nose tests
def setup_module(module):
from nose import SkipTest
try:
import numpy
except:
raise SkipTest("NumPy not available")
try:
import scipy
except:
raise SkipTest("SciPy not available")