"""
******
Layout
******
Node positioning algorithms for graph drawing.
The default scales and centering for these layouts are
typically squares with side [0, 1] or [0, scale].
The two circular layout routines (circular_layout and
shell_layout) have size [-1, 1] or [-scale, scale].
"""
# Authors: Aric Hagberg <aric.hagberg@gmail.com>,
# Dan Schult <dschult@colgate.edu>
# 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.
import collections
import networkx as nx
__all__ = ['circular_layout',
'random_layout',
'shell_layout',
'spring_layout',
'spectral_layout',
'fruchterman_reingold_layout']
[docs]def random_layout(G, dim=2, scale=1., center=None):
"""Position nodes uniformly at random.
For every node, a position is generated by choosing each of dim
coordinates uniformly at random on the default interval [0.0, 1.0),
or on an interval of length `scale` centered at `center`.
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.
dim : int
Dimension of layout.
scale : float (default 1)
Scale factor for positions
center : array-like (default scale*0.5 in each dim)
Coordinate around which to center the 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
shape = (len(G), dim)
pos = np.random.random(shape) * scale
if center is not None:
pos += np.asarray(center) - 0.5 * scale
return dict(zip(G, pos))
[docs]def circular_layout(G, dim=2, scale=1., center=None):
"""Position nodes on a circle.
Parameters
----------
G : NetworkX graph or list of nodes
dim : int
Dimension of layout, currently only dim=2 is supported
scale : float (default 1)
Scale factor for positions, i.e. radius of circle.
center : array-like (default origin)
Coordinate around which to center the layout.
Returns
-------
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
if len(G) == 0:
return {}
twopi = 2.0*np.pi
theta = np.arange(0, twopi, twopi/len(G))
pos = np.column_stack([np.cos(theta), np.sin(theta)]) * scale
if center is not None:
pos += np.asarray(center)
return dict(zip(G, pos))
[docs]def shell_layout(G, nlist=None, dim=2, scale=1., center=None):
"""Position nodes in concentric circles.
Parameters
----------
G : NetworkX graph or list of nodes
nlist : list of lists
List of node lists for each shell.
dim : int
Dimension of layout, currently only dim=2 is supported
scale : float (default 1)
Scale factor for positions, i.e.radius of largest shell
center : array-like (default origin)
Coordinate around which to center the layout.
Returns
-------
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
if len(G) == 0:
return {}
if nlist is None:
# draw the whole graph in one shell
nlist = [list(G)]
numb_shells = len(nlist)
if len(nlist[0]) == 1:
# single node at center
radius = 0.0
numb_shells -= 1
else:
# else start at r=1
radius = 1.0
# distance between shells
gap = (scale / numb_shells) if numb_shells else scale
radius *= gap
npos={}
twopi = 2.0*np.pi
for nodes in nlist:
theta = np.arange(0, twopi, twopi/len(nodes))
pos = np.column_stack([np.cos(theta), np.sin(theta)]) * radius
npos.update(zip(nodes, pos))
radius += gap
if center is not None:
center = np.asarray(center)
for n,p in npos.items():
npos[n] = p + center
return npos
[docs]def fruchterman_reingold_layout(G, dim=2, k=None,
pos=None,
fixed=None,
iterations=50,
weight='weight',
scale=1.0,
center=None):
"""Position nodes using Fruchterman-Reingold force-directed algorithm.
Parameters
----------
G : NetworkX graph
dim : int
Dimension of layout
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 list or tuple. If None, then use random initial
positions.
fixed : list or None optional (default=None)
Nodes to keep fixed at initial position.
If any nodes are fixed, the scale and center features are not used.
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 effective spring constant. If None, edge weights are 1.
scale : float (default=1.0)
Scale factor for positions. The nodes are positioned
in a box of size `scale` in each dim centered at `center`.
center : array-like (default scale/2 in each dim)
Coordinate around which to center the layout.
Returns
-------
dict :
A dictionary of positions keyed by node
Examples
--------
>>> G=nx.path_graph(4)
>>> pos=nx.spring_layout(G)
# this function has two names:
# spring_layout and fruchterman_reingold_layout
>>> pos=nx.fruchterman_reingold_layout(G)
"""
import numpy as np
if len(G) == 0:
return {}
if fixed is not None:
nfixed = dict(zip(G, range(len(G))))
fixed = np.asarray([nfixed[v] for v in fixed])
if pos is None:
msg = "Keyword pos must be specified if any nodes are fixed"
raise ValueError(msg)
if pos is not None:
# Determine size of existing domain to adjust initial positions
pos_coords = np.array(list(pos.values()))
min_coords = pos_coords.min(0)
domain_size = pos_coords.max(0) - min_coords
shape = (len(G), dim)
pos_arr = np.random.random(shape) * domain_size + min_coords
for i,n in enumerate(G):
if n in pos:
pos_arr[i] = np.asarray(pos[n])
else:
pos_arr=None
if k is None and fixed is not None:
# Adjust k for domains larger than 1x1
k=domain_size.max()/np.sqrt(len(G))
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')
pos = _sparse_fruchterman_reingold(A,dim,k,pos_arr,fixed,iterations)
except:
A = nx.to_numpy_matrix(G, weight=weight)
pos = _fruchterman_reingold(A, dim, k, pos_arr, fixed, iterations)
if fixed is None:
pos = _rescale_layout(pos, scale)
if center is not None:
pos += np.asarray(center) - 0.5 * scale
return dict(zip(G,pos))
spring_layout=fruchterman_reingold_layout
def _fruchterman_reingold(A,dim=2,k=None,pos=None,fixed=None,iterations=50):
# Position nodes in adjacency matrix A using Fruchterman-Reingold
# Entry point for NetworkX graph is fruchterman_reingold_layout()
import numpy as np
try:
nnodes,_=A.shape
except AttributeError:
raise nx.NetworkXError(
"fruchterman_reingold() takes an adjacency matrix as input")
A=np.asarray(A) # make sure we have an array instead of a matrix
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.
# Calculate domain in case our fixed positions are 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<0.01,0.01,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
if fixed is None:
pos = _rescale_layout(pos)
return pos
def _sparse_fruchterman_reingold(A, dim=2, k=None, pos=None, fixed=None,
iterations=50):
# Position nodes in adjacency matrix A using Fruchterman-Reingold
# Entry point for NetworkX graph is fruchterman_reingold_layout()
# Sparse version
import numpy as np
try:
nnodes,_=A.shape
except AttributeError:
raise nx.NetworkXError(
"fruchterman_reingold() takes an adjacency matrix as input")
try:
from scipy.sparse import spdiags,coo_matrix
except ImportError:
raise ImportError("_sparse_fruchterman_reingold() scipy numpy: http://scipy.org/ ")
# 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.
# Calculate domain in case our fixed positions are 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)
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<0.01,0.01,length)
pos+=(displacement*t/length).T
# cool temperature
t-=dt
if fixed is None:
pos = _rescale_layout(pos)
return pos
[docs]def spectral_layout(G, dim=2, weight='weight', scale=1., center=None):
"""Position nodes using the eigenvectors of the graph Laplacian.
Parameters
----------
G : NetworkX graph or list of nodes
dim : int
Dimension of layout
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 optional (default 1)
Scale factor for positions, i.e. nodes placed in a box with
side [0, scale] or centered on `center` if provided.
center : array-like (default scale/2 in each dim)
Coordinate around which to center the layout.
Returns
-------
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
if len(G) <= 2:
if len(G) == 0:
return {}
elif len(G) == 1:
if center is not None:
pos = np.asarray(center)
else:
pos = np.ones((1,dim)) * scale * 0.5
else: #len(G) == 2
pos = np.array([np.zeros(dim), np.ones(dim) * scale])
if center is not None:
pos += np.asarray(center) - scale * 0.5
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)
if center is not None:
pos += np.asarray(center) - 0.5 * scale
return dict(zip(G,pos))
def _spectral(A, dim=2):
# Input adjacency matrix A
# Uses dense eigenvalue solver from numpy
try:
import numpy as np
except ImportError:
raise ImportError("spectral_layout() requires numpy: http://scipy.org/ ")
try:
nnodes,_=A.shape
except AttributeError:
raise nx.NetworkXError(\
"spectral() takes an adjacency matrix as input")
# 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:
raise ImportError("_sparse_spectral() requires scipy & numpy: http://scipy.org/ ")
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:
raise nx.NetworkXError(\
"sparse_spectral() takes an adjacency matrix as input")
# 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])
def _rescale_layout(pos, scale=1.):
# rescale to [0, scale) in each axis
# Find max length over all dimensions
maxlim=0
for i in range(pos.shape[1]):
pos[:,i] -= pos[:,i].min() # shift min to zero
maxlim = max(maxlim, pos[:,i].max())
if maxlim > 0:
for i in range(pos.shape[1]):
pos[:,i] *= scale / maxlim
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")