Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.drawing.layout

"""
******
Layout
******

Node positioning algorithms for graph drawing.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
import networkx as nx
__author__ = """Aric Hagberg (hagberg@lanl.gov)\nDan Schult(dschult@colgate.edu)"""
__all__ = ['circular_layout',
'random_layout',
'shell_layout',
'spring_layout',
'spectral_layout',
'fruchterman_reingold_layout']

[docs]def random_layout(G,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
A position will be assigned to every node in G.

dim : int
Dimension of layout.

Returns
-------
dict :
A dictionary of positions keyed by node

Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> pos = nx.random_layout(G)

"""
try:
import numpy as np
except ImportError:
raise ImportError("random_layout() requires numpy: http://scipy.org/ ")
n=len(G)
pos=np.asarray(np.random.random((n,dim)),dtype=np.float32)
return dict(zip(G,pos))

[docs]def circular_layout(G, dim=2, scale=1):
# dim=2 only
"""Position nodes on a circle.

Parameters
----------
G : NetworkX graph

dim : int
Dimension of layout, currently only dim=2 is supported

scale : float
Scale factor for positions

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.

"""
try:
import numpy as np
except ImportError:
raise ImportError("circular_layout() requires numpy: http://scipy.org/ ")
if len(G)==0:
return {}
if len(G)==1:
return {G.nodes()[0]:(1,)*dim}
t=np.arange(0,2.0*np.pi,2.0*np.pi/len(G),dtype=np.float32)
pos=np.transpose(np.array([np.cos(t),np.sin(t)]))
pos=_rescale_layout(pos,scale=scale)
return dict(zip(G,pos))

[docs]def shell_layout(G,nlist=None,dim=2,scale=1):
"""Position nodes in concentric circles.

Parameters
----------
G : NetworkX graph

nlist : list of lists
List of node lists for each shell.

dim : int
Dimension of layout, currently only dim=2 is supported

scale : float
Scale factor for positions

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.

"""
try:
import numpy as np
except ImportError:
raise ImportError("shell_layout() requires numpy: http://scipy.org/ ")
if len(G)==0:
return {}
if len(G)==1:
return {G.nodes()[0]:(1,)*dim}
if nlist==None:
nlist=[G.nodes()] # draw the whole graph in one shell

if len(nlist[0])==1:
radius=0.0 # single node at center
else:
radius=1.0 # else start at r=1

npos={}
for nodes in nlist:
t=np.arange(0,2.0*np.pi,2.0*np.pi/len(nodes),dtype=np.float32)
npos.update(zip(nodes,pos))

# FIXME: rescale
return npos

def fruchterman_reingold_layout(G,dim=2,k=None,
pos=None,
fixed=None,
iterations=50,
weight='weight',
scale=1.0):
"""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 nuse 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].

Returns
-------
dict :
A dictionary of positions keyed by node

Examples
--------
>>> G=nx.path_graph(4)
>>> pos=nx.spring_layout(G)

# The same using longer function name
>>> pos=nx.fruchterman_reingold_layout(G)
"""
try:
import numpy as np
except ImportError:
raise ImportError("fruchterman_reingold_layout() requires numpy: http://scipy.org/ ")
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:
pos_arr=np.asarray(np.random.random((len(G),dim)))
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 {G.nodes()[0]:(1,)*dim}

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=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()
try:
import numpy as np
except ImportError:
raise ImportError("_fruchterman_reingold() requires numpy: http://scipy.org/ ")

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==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.
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)
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
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
try:
import numpy as np
except ImportError:
raise ImportError("_sparse_fruchterman_reingold() requires numpy: http://scipy.org/ ")
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==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==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)
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
pos=_rescale_layout(pos)
return pos

[docs]def spectral_layout(G, dim=2, weight='weight', scale=1):
"""Position nodes using the eigenvectors of the graph Laplacian.

Parameters
----------
G : NetworkX graph

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
Scale factor for positions

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 unidrected 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
try:
import numpy as np
except ImportError:
raise ImportError("spectral_layout() requires numpy: http://scipy.org/ ")
if len(G)<=2:
if len(G)==0:
pos=np.array([])
elif len(G)==1:
pos=np.array([[1,1]])
else:
pos=np.array([[0,0.5],[1,0.5]])
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)
return dict(zip(G,pos))

def _spectral(A, dim=2):
# 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):
# 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,pscale) in all axes

# shift origin to (0,0)
lim=0 # max coordinate for all axes
for i in range(pos.shape[1]):
pos[:,i]-=pos[:,i].min()
lim=max(pos[:,i].max(),lim)
# rescale to (0,scale) in all directions, preserves aspect
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")