# Source code for networkx.drawing.layout

```"""
******
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.

"""
import networkx as nx
from networkx.utils import np_random_state

__all__ = [
"bipartite_layout",
"circular_layout",
"random_layout",
"rescale_layout",
"rescale_layout_dict",
"shell_layout",
"spring_layout",
"spectral_layout",
"planar_layout",
"fruchterman_reingold_layout",
"spiral_layout",
"multipartite_layout",
"arf_layout",
]

def _process_params(G, center, dim):
# Some boilerplate code.
import numpy as np

if not isinstance(G, nx.Graph):
empty_graph = nx.Graph()
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]@np_random_state(3)
def random_layout(G, center=None, dim=2, seed=None):
"""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.

seed : int, RandomState instance or None  optional (default=None)
Set the random state for deterministic node layouts.
If int, `seed` is the seed used by the random number generator,
if numpy.random.RandomState instance, `seed` is the random
number generator,
if None, the random number generator is the RandomState instance used
by numpy.random.

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)
pos = seed.rand(len(G), dim) + 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
A position will be assigned to every node in G.

scale : number (default: 1)
Scale factor for positions.

center : array-like or None
Coordinate pair around which to center the layout.

dim : int
Dimension of layout.
If dim>2, the remaining dimensions are set to zero
in the returned positions.
If dim<2, a ValueError is raised.

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

Raises
------
ValueError
If dim < 2

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 dim < 2:
raise ValueError("cannot handle dimensions < 2")

G, center = _process_params(G, center, dim)

paddims = max(0, (dim - 2))

if len(G) == 0:
pos = {}
elif len(G) == 1:
pos = {nx.utils.arbitrary_element(G): center}
else:
theta = np.linspace(0, 1, len(G) + 1)[:-1] * 2 * np.pi
theta = theta.astype(np.float32)
pos = np.column_stack(
)
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))

return pos

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

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

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

rotate : angle in radians (default=pi/len(nlist))
Angle by which to rotate the starting position of each shell
relative to the starting position of the previous shell.
To recreate behavior before v2.5 use rotate=0.

scale : number (default: 1)
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.
Other dimension values result in a ValueError.

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

Raises
------
ValueError
If dim != 2

Examples
--------
>>> G = nx.path_graph(4)
>>> shells = [, [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 dim != 2:
raise ValueError("can only handle 2 dimensions")

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) == 1:
# single node at center
else:
# else start at r=1

if rotate is None:
rotate = np.pi / len(nlist)
first_theta = rotate
npos = {}
for nodes in nlist:
theta = (
np.linspace(0, 2 * np.pi, len(nodes), endpoint=False, dtype=np.float32)
+ first_theta
)
pos = radius * np.column_stack([np.cos(theta), np.sin(theta)]) + center
npos.update(zip(nodes, pos))
first_theta += rotate

return npos

[docs]def bipartite_layout(
G, nodes, align="vertical", scale=1, center=None, aspect_ratio=4 / 3
):
"""Position nodes in two straight lines.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

nodes : list or container
Nodes in one node set of the bipartite graph.
This set will be placed on left or top.

align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.

scale : number (default: 1)
Scale factor for positions.

center : array-like or None
Coordinate pair around which to center the layout.

aspect_ratio : number (default=4/3):
The ratio of the width to the height of the layout.

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

Examples
--------
>>> G = nx.bipartite.gnmk_random_graph(3, 5, 10, seed=123)
>>> top = nx.bipartite.sets(G)
>>> pos = nx.bipartite_layout(G, top)

Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.

"""

import numpy as np

if align not in ("vertical", "horizontal"):
msg = "align must be either vertical or horizontal."
raise ValueError(msg)

G, center = _process_params(G, center=center, dim=2)
if len(G) == 0:
return {}

height = 1
width = aspect_ratio * height
offset = (width / 2, height / 2)

top = set(nodes)
bottom = set(G) - top
nodes = list(top) + list(bottom)

left_xs = np.repeat(0, len(top))
right_xs = np.repeat(width, len(bottom))
left_ys = np.linspace(0, height, len(top))
right_ys = np.linspace(0, height, len(bottom))

top_pos = np.column_stack([left_xs, left_ys]) - offset
bottom_pos = np.column_stack([right_xs, right_ys]) - offset

pos = np.concatenate([top_pos, bottom_pos])
pos = rescale_layout(pos, scale=scale) + center
if align == "horizontal":
pos = pos[:, ::-1]  # swap x and y coords
pos = dict(zip(nodes, pos))
return pos

[docs]@np_random_state(10)
def spring_layout(
G,
k=None,
pos=None,
fixed=None,
iterations=50,
threshold=1e-4,
weight="weight",
scale=1,
center=None,
dim=2,
seed=None,
):
"""Position nodes using Fruchterman-Reingold force-directed algorithm.

The algorithm simulates a force-directed representation of the network
treating edges as springs holding nodes close, while treating nodes
as repelling objects, sometimes called an anti-gravity force.
Simulation continues until the positions are close to an equilibrium.

There are some hard-coded values: minimal distance between
nodes (0.01) and "temperature" of 0.1 to ensure nodes don't fly away.
During the simulation, `k` helps determine the distance between nodes,
though `scale` and `center` determine the size and place after
rescaling occurs at the end of the simulation.

Fixing some nodes doesn't allow them to move in the simulation.
It also turns off the rescaling feature at the simulation's end.
In addition, setting `scale` to `None` turns off rescaling.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

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.
Nodes not in ``G.nodes`` are ignored.
ValueError raised if `fixed` specified and `pos` not.

iterations : int  optional (default=50)
Maximum number of iterations taken

threshold: float optional (default = 1e-4)
Threshold for relative error in node position changes.
The iteration stops if the error is below this threshold.

weight : string or None   optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight.  Larger means a stronger attractive force.
If None, then all edge weights are 1.

scale : number or None (default: 1)
Scale factor for positions. Not used unless `fixed is None`.
If scale is None, no rescaling is performed.

center : array-like or None
Coordinate pair around which to center the layout.
Not used unless `fixed is None`.

dim : int
Dimension of layout.

seed : int, RandomState instance or None  optional (default=None)
Set the random state for deterministic node layouts.
If int, `seed` is the seed used by the random number generator,
if numpy.random.RandomState instance, `seed` is the random
number generator,
if None, the random number generator is the RandomState instance used
by numpy.random.

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:
if pos is None:
raise ValueError("nodes are fixed without positions given")
for node in fixed:
if node not in pos:
raise ValueError("nodes are fixed without positions given")
nfixed = {node: i for i, node in enumerate(G)}
fixed = np.asarray([nfixed[node] for node in fixed if node in nfixed])

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)
if dom_size == 0:
dom_size = 1
pos_arr = seed.rand(len(G), dim) * dom_size + center

for i, n in enumerate(G):
if n in pos:
pos_arr[i] = np.asarray(pos[n])
else:
pos_arr = None
dom_size = 1

if len(G) == 0:
return {}
if len(G) == 1:
return {nx.utils.arbitrary_element(G.nodes()): center}

try:
# Sparse matrix
if len(G) < 500:  # sparse solver for large graphs
raise ValueError
A = nx.to_scipy_sparse_array(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, threshold, dim, seed
)
except ValueError:
A = nx.to_numpy_array(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, threshold, dim, seed
)
if fixed is None and scale is not None:
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos

fruchterman_reingold_layout = spring_layout

@np_random_state(7)
def _fruchterman_reingold(
A, k=None, pos=None, fixed=None, iterations=50, threshold=1e-4, dim=2, seed=None
):
# 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 as err:
msg = "fruchterman_reingold() takes an adjacency matrix as input"
raise nx.NetworkXError(msg) from err

if pos is None:
# random initial positions
pos = np.asarray(seed.rand(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) - min(pos.T), max(pos.T) - min(pos.T)) * 0.1
# simple cooling scheme.
# linearly step down by dt on each iteration so last iteration is size dt.
dt = t / (iterations + 1)
delta = np.zeros((pos.shape, pos.shape, pos.shape), 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
delta = pos[:, np.newaxis, :] - pos[np.newaxis, :, :]
# distance between points
distance = np.linalg.norm(delta, axis=-1)
# enforce minimum distance of 0.01
np.clip(distance, 0.01, None, out=distance)
# displacement "force"
displacement = np.einsum(
"ijk,ij->ik", delta, (k * k / distance**2 - A * distance / k)
)
# update positions
length = np.linalg.norm(displacement, axis=-1)
length = np.where(length < 0.01, 0.1, length)
delta_pos = np.einsum("ij,i->ij", 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 (np.linalg.norm(delta_pos) / nnodes) < threshold:
break
return pos

@np_random_state(7)
def _sparse_fruchterman_reingold(
A, k=None, pos=None, fixed=None, iterations=50, threshold=1e-4, dim=2, seed=None
):
# Position nodes in adjacency matrix A using Fruchterman-Reingold
# Entry point for NetworkX graph is fruchterman_reingold_layout()
# Sparse version
import numpy as np
import scipy as sp
import scipy.sparse  # call as sp.sparse

try:
nnodes, _ = A.shape
except AttributeError as err:
msg = "fruchterman_reingold() takes an adjacency matrix as input"
raise nx.NetworkXError(msg) from err
# make sure we have a LIst of Lists representation
try:
A = A.tolil()
except AttributeError:
A = (sp.sparse.coo_array(A)).tolil()

if pos is None:
# random initial positions
pos = np.asarray(seed.rand(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 = max(max(pos.T) - min(pos.T), max(pos.T) - min(pos.T)) * 0.1
# simple cooling scheme.
# linearly step down by dt on each iteration so last iteration is size dt.
dt = t / (iterations + 1)

displacement = np.zeros((dim, nnodes))
for iteration in range(iterations):
displacement *= 0
# loop over rows
for i in range(A.shape):
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 = A.getrowview(i).toarray()  # TODO: revisit w/ sparse 1D container
# 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)
delta_pos = (displacement * t / length).T
pos += delta_pos
# cool temperature
t -= dt
if (np.linalg.norm(delta_pos) / nnodes) < threshold:
break
return pos

G, dist=None, pos=None, weight="weight", scale=1, center=None, dim=2
):
"""Position nodes using Kamada-Kawai path-length cost-function.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

dist : dict (default=None)
A two-level dictionary of optimal distances between nodes,
indexed by source and destination node.
If None, the distance is computed using shortest_path_length().

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
circular_layout() for dim >= 2 and a linear layout for dim == 1.

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 : number (default: 1)
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)
"""
import numpy as np

G, center = _process_params(G, center, dim)
nNodes = len(G)
if nNodes == 0:
return {}

if dist is None:
dist = dict(nx.shortest_path_length(G, weight=weight))
dist_mtx = 1e6 * np.ones((nNodes, nNodes))
for row, nr in enumerate(G):
if nr not in dist:
continue
rdist = dist[nr]
for col, nc in enumerate(G):
if nc not in rdist:
continue
dist_mtx[row][col] = rdist[nc]

if pos is None:
if dim >= 3:
pos = random_layout(G, dim=dim)
elif dim == 2:
pos = circular_layout(G, dim=dim)
else:
pos = dict(zip(G, np.linspace(0, 1, len(G))))
pos_arr = np.array([pos[n] for n in G])

pos = rescale_layout(pos, scale=scale) + center
return dict(zip(G, pos))

# Anneal node locations based on the Kamada-Kawai cost-function,
# using the supplied matrix of preferred inter-node distances,
# and starting locations.

import numpy as np
import scipy as sp
import scipy.optimize  # call as sp.optimize

meanwt = 1e-3
costargs = (np, 1 / (dist_mtx + np.eye(dist_mtx.shape) * 1e-3), meanwt, dim)

optresult = sp.optimize.minimize(
pos_arr.ravel(),
method="L-BFGS-B",
args=costargs,
jac=True,
)

return optresult.x.reshape((-1, dim))

def _kamada_kawai_costfn(pos_vec, np, invdist, meanweight, dim):
nNodes = invdist.shape
pos_arr = pos_vec.reshape((nNodes, dim))

delta = pos_arr[:, np.newaxis, :] - pos_arr[np.newaxis, :, :]
nodesep = np.linalg.norm(delta, axis=-1)
direction = np.einsum("ijk,ij->ijk", delta, 1 / (nodesep + np.eye(nNodes) * 1e-3))

offset = nodesep * invdist - 1.0
offset[np.diag_indices(nNodes)] = 0

cost = 0.5 * np.sum(offset**2)
grad = np.einsum("ij,ij,ijk->ik", invdist, offset, direction) - np.einsum(
"ij,ij,ijk->jk", invdist, offset, direction
)

# Additional parabolic term to encourage mean position to be near origin:
sumpos = np.sum(pos_arr, axis=0)
cost += 0.5 * meanweight * np.sum(sumpos**2)

[docs]def spectral_layout(G, weight="weight", scale=1, center=None, dim=2):
"""Position nodes using the eigenvectors of the graph Laplacian.

Using the unnormalized Laplacian, the layout shows possible clusters of
nodes which are an approximation of the ratio cut. If dim is the number of
dimensions then the positions are the entries of the dim eigenvectors
corresponding to the ascending eigenvalues starting from the second one.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

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 : number (default: 1)
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_array(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_array(G, weight=weight)
# Symmetrize directed graphs
if G.is_directed():
A += A.T
pos = _spectral(A, dim)

pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos

def _spectral(A, dim=2):
# Uses dense eigenvalue solver from numpy
import numpy as np

try:
nnodes, _ = A.shape
except AttributeError as err:
msg = "spectral() takes an adjacency matrix as input"
raise nx.NetworkXError(msg) from err

# form Laplacian matrix where D is diagonal of degrees
D = np.identity(nnodes, dtype=A.dtype) * np.sum(A, axis=1)
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"
import numpy as np
import scipy as sp
import scipy.sparse  # call as sp.sparse
import scipy.sparse.linalg  # call as sp.sparse.linalg

try:
nnodes, _ = A.shape
except AttributeError as err:
msg = "sparse_spectral() takes an adjacency matrix as input"
raise nx.NetworkXError(msg) from err

# form Laplacian matrix
# TODO: Rm csr_array wrapper in favor of spdiags array constructor when available
D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 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 = sp.sparse.linalg.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 planar_layout(G, scale=1, center=None, dim=2):
"""Position nodes without edge intersections.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G. If G is of type
nx.PlanarEmbedding, the positions are selected accordingly.

scale : number (default: 1)
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

Raises
------
NetworkXException
If G is not planar

Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.planar_layout(G)
"""
import numpy as np

if dim != 2:
raise ValueError("can only handle 2 dimensions")

G, center = _process_params(G, center, dim)

if len(G) == 0:
return {}

if isinstance(G, nx.PlanarEmbedding):
embedding = G
else:
is_planar, embedding = nx.check_planarity(G)
if not is_planar:
raise nx.NetworkXException("G is not planar.")
pos = nx.combinatorial_embedding_to_pos(embedding)
node_list = list(embedding)
pos = np.row_stack([pos[x] for x in node_list])
pos = pos.astype(np.float64)
pos = rescale_layout(pos, scale=scale) + center
return dict(zip(node_list, pos))

[docs]def spiral_layout(G, scale=1, center=None, dim=2, resolution=0.35, equidistant=False):
"""Position nodes in a spiral layout.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int, default=2
Dimension of layout, currently only dim=2 is supported.
Other dimension values result in a ValueError.
resolution : float, default=0.35
The compactness of the spiral layout returned.
Lower values result in more compressed spiral layouts.
equidistant : bool, default=False
If True, nodes will be positioned equidistant from each other
by decreasing angle further from center.
If False, nodes will be positioned at equal angles
from each other by increasing separation further from center.

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

Raises
------
ValueError
If dim != 2

Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.spiral_layout(G)
>>> nx.draw(G, pos=pos)

Notes
-----
This algorithm currently only works in two dimensions.

"""
import numpy as np

if dim != 2:
raise ValueError("can only handle 2 dimensions")

G, center = _process_params(G, center, dim)

if len(G) == 0:
return {}
if len(G) == 1:
return {nx.utils.arbitrary_element(G): center}

pos = []
if equidistant:
chord = 1
step = 0.5
theta = resolution
theta += chord / (step * theta)
for _ in range(len(G)):
r = step * theta
theta += chord / r
pos.append([np.cos(theta) * r, np.sin(theta) * r])

else:
dist = np.arange(len(G), dtype=float)
angle = resolution * dist
pos = np.transpose(dist * np.array([np.cos(angle), np.sin(angle)]))

pos = rescale_layout(np.array(pos), scale=scale) + center

pos = dict(zip(G, pos))

return pos

[docs]def multipartite_layout(G, subset_key="subset", align="vertical", scale=1, center=None):
"""Position nodes in layers of straight lines.

Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.

subset_key : string (default='subset')
Key of node data to be used as layer subset.

align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.

scale : number (default: 1)
Scale factor for positions.

center : array-like or None
Coordinate pair around which to center the layout.

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

Examples
--------
>>> G = nx.complete_multipartite_graph(28, 16, 10)
>>> pos = nx.multipartite_layout(G)

Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.

Network does not need to be a complete multipartite graph. As long as nodes
have subset_key data, they will be placed in the corresponding layers.

"""
import numpy as np

if align not in ("vertical", "horizontal"):
msg = "align must be either vertical or horizontal."
raise ValueError(msg)

G, center = _process_params(G, center=center, dim=2)
if len(G) == 0:
return {}

layers = {}
for v, data in G.nodes(data=True):
try:
layer = data[subset_key]
except KeyError:
msg = "all nodes must have subset_key (default='subset') as data"
raise ValueError(msg)
layers[layer] = [v] + layers.get(layer, [])

# Sort by layer, if possible
try:
layers = sorted(layers.items())
except TypeError:
layers = list(layers.items())

pos = None
nodes = []
width = len(layers)
for i, (_, layer) in enumerate(layers):
height = len(layer)
xs = np.repeat(i, height)
ys = np.arange(0, height, dtype=float)
offset = ((width - 1) / 2, (height - 1) / 2)
layer_pos = np.column_stack([xs, ys]) - offset
if pos is None:
pos = layer_pos
else:
pos = np.concatenate([pos, layer_pos])
nodes.extend(layer)
pos = rescale_layout(pos, scale=scale) + center
if align == "horizontal":
pos = pos[:, ::-1]  # swap x and y coords
pos = dict(zip(nodes, pos))
return pos

def arf_layout(
G,
pos=None,
scaling=1,
a=1.1,
etol=1e-6,
dt=1e-3,
max_iter=1000,
):
"""Arf layout for networkx

The attractive and repulsive forces (arf) layout 
improves the spring layout in three ways. First, it
prevents congestion of highly connected nodes due to
strong forcing between nodes. Second, it utilizes the
layout space more effectively by preventing large gaps
that spring layout tends to create. Lastly, the arf
layout represents symmmetries in the layout better than
the default spring layout.

Parameters
----------
G : nx.Graph or nx.DiGraph
Networkx graph.
pos : dict
Initial  position of  the nodes.  If set  to None  a
random layout will be used.
scaling : float
Scales the radius of the circular layout space.
a : float
Strength of springs between connected nodes. Should be larger than 1. The greater a, the clearer the separation ofunconnected sub clusters.
etol : float
Graduent sum of spring forces must be larger than `etol` before successful termination.
dt : float
Time step for force differential equation simulations.
max_iter : int
Max iterations before termination of the algorithm.

References
..  "Self-Organization Applied to Dynamic Network Layout", M. Geipel,
International Journal of Modern Physics C, 2007, Vol 18, No 10, pp. 1537-1549.
https://doi.org/10.1142/S0129183107011558 https://arxiv.org/abs/0704.1748

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

Examples
--------
>>> G = nx.grid_graph((5, 5))
>>> pos = nx.arf_layout(G)

"""
import warnings

import numpy as np

if a <= 1:
msg = "The parameter a should be larger than 1"
raise ValueError(msg)

pos_tmp = nx.random_layout(G)
if pos is None:
pos = pos_tmp
else:
for node in G.nodes():
if node not in pos:
pos[node] = pos_tmp[node].copy()

# Initialize spring constant matrix
N = len(G)
# No nodes no computation
if N == 0:
return pos

# init force of springs
K = np.ones((N, N)) - np.eye(N)
node_order = {node: i for i, node in enumerate(G)}
for x, y in G.edges():
if x != y:
idx, jdx = (node_order[i] for i in (x, y))
K[idx, jdx] = a

# vectorize values
p = np.asarray(list(pos.values()))

# equation 10 in 
rho = scaling * np.sqrt(N)

# looping variables
error = etol + 1
n_iter = 0
while error > etol:
diff = p[:, np.newaxis] - p[np.newaxis]
A = np.linalg.norm(diff, axis=-1)[..., np.newaxis]
# attraction_force - repulsions force
# suppress nans due to division; caused by diagonal set to zero.
# Does not affect the computation due to nansum
with warnings.catch_warnings():
warnings.simplefilter("ignore")
change = K[..., np.newaxis] * diff - rho / A * diff
change = np.nansum(change, axis=0)
p += change * dt

error = np.linalg.norm(change, axis=-1).sum()
if n_iter > max_iter:
break
n_iter += 1
return dict(zip(G.nodes(), p))

[docs]def rescale_layout(pos, scale=1):
"""Returns 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.

--------
rescale_layout_dict
"""
# Find max length over all dimensions
lim = 0  # max coordinate for all axes
for i in range(pos.shape):
pos[:, i] -= pos[:, i].mean()
lim = max(abs(pos[:, i]).max(), lim)
# rescale to (-scale, scale) in all directions, preserves aspect
if lim > 0:
for i in range(pos.shape):
pos[:, i] *= scale / lim
return pos

[docs]def rescale_layout_dict(pos, scale=1):
"""Return a dictionary of scaled positions keyed by node

Parameters
----------
pos : A dictionary of positions keyed by node

scale : number (default: 1)
The size of the resulting extent in all directions.

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

Examples
--------
>>> import numpy as np
>>> pos = {0: np.array((0, 0)), 1: np.array((1, 1)), 2: np.array((0.5, 0.5))}
>>> nx.rescale_layout_dict(pos)
{0: array([-1., -1.]), 1: array([1., 1.]), 2: array([0., 0.])}

>>> pos = {0: np.array((0, 0)), 1: np.array((-1, 1)), 2: np.array((-0.5, 0.5))}
>>> nx.rescale_layout_dict(pos, scale=2)
{0: array([ 2., -2.]), 1: array([-2.,  2.]), 2: array([0., 0.])}