1184 lines
35 KiB
Python
1184 lines
35 KiB
Python
"""
|
|
******
|
|
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",
|
|
"kamada_kawai_layout",
|
|
"random_layout",
|
|
"rescale_layout",
|
|
"rescale_layout_dict",
|
|
"shell_layout",
|
|
"spring_layout",
|
|
"spectral_layout",
|
|
"planar_layout",
|
|
"fruchterman_reingold_layout",
|
|
"spiral_layout",
|
|
"multipartite_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
|
|
|
|
|
|
@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
|
|
|
|
|
|
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:
|
|
# 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), np.zeros((len(G), paddims))]
|
|
)
|
|
pos = rescale_layout(pos, scale=scale) + center
|
|
pos = dict(zip(G, pos))
|
|
|
|
return pos
|
|
|
|
|
|
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 = [[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 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)]
|
|
|
|
radius_bump = scale / len(nlist)
|
|
|
|
if len(nlist[0]) == 1:
|
|
# single node at center
|
|
radius = 0.0
|
|
else:
|
|
# else start at r=1
|
|
radius = radius_bump
|
|
|
|
if rotate is None:
|
|
rotate = np.pi / len(nlist)
|
|
first_theta = rotate
|
|
npos = {}
|
|
for nodes in nlist:
|
|
# Discard the last angle (endpoint=False) since 2*pi matches 0 radians
|
|
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))
|
|
radius += radius_bump
|
|
first_theta += rotate
|
|
|
|
return npos
|
|
|
|
|
|
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)[0]
|
|
>>> 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 = np.flip(pos, 1)
|
|
pos = dict(zip(nodes, pos))
|
|
return pos
|
|
|
|
|
|
@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[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
|
|
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[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 = 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
|
|
|
|
|
|
def kamada_kawai_layout(
|
|
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)
|
|
>>> pos = nx.kamada_kawai_layout(G)
|
|
"""
|
|
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 = {n: pt for n, pt in zip(G, np.linspace(0, 1, len(G)))}
|
|
pos_arr = np.array([pos[n] for n in G])
|
|
|
|
pos = _kamada_kawai_solve(dist_mtx, pos_arr, dim)
|
|
|
|
pos = rescale_layout(pos, scale=scale) + center
|
|
return dict(zip(G, pos))
|
|
|
|
|
|
def _kamada_kawai_solve(dist_mtx, pos_arr, dim):
|
|
# 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[0]) * 1e-3), meanwt, dim)
|
|
|
|
optresult = sp.optimize.minimize(
|
|
_kamada_kawai_costfn,
|
|
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):
|
|
# Cost-function and gradient for Kamada-Kawai layout algorithm
|
|
nNodes = invdist.shape[0]
|
|
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)
|
|
grad += meanweight * sumpos
|
|
|
|
return (cost, grad.ravel())
|
|
|
|
|
|
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):
|
|
# Input adjacency matrix A
|
|
# 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):
|
|
# Input adjacency matrix A
|
|
# 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])
|
|
|
|
|
|
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))
|
|
|
|
|
|
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)
|
|
|
|
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
|
|
|
|
|
|
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, [])
|
|
|
|
pos = None
|
|
nodes = []
|
|
|
|
width = len(layers)
|
|
for i, layer in enumerate(layers.values()):
|
|
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 = np.flip(pos, 1)
|
|
pos = dict(zip(nodes, pos))
|
|
return pos
|
|
|
|
|
|
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.
|
|
|
|
See Also
|
|
--------
|
|
rescale_layout_dict
|
|
"""
|
|
# 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(abs(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
|
|
|
|
|
|
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.])}
|
|
|
|
See Also
|
|
--------
|
|
rescale_layout
|
|
"""
|
|
import numpy as np
|
|
|
|
if not pos: # empty_graph
|
|
return {}
|
|
pos_v = np.array(list(pos.values()))
|
|
pos_v = rescale_layout(pos_v, scale=scale)
|
|
return dict(zip(pos, pos_v))
|