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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

#    Copyright (C) 2004-2017 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    Richard Penney <rwpenney@users.sourceforge.net>
#    All rights reserved.
#    BSD license.
#
# Authors: Aric Hagberg <aric.hagberg@gmail.com>,
#          Dan Schult <dschult@colgate.edu>
"""
******
Layout
******

Node positioning algorithms for graph drawing.

For `random_layout()` the possible resulting shape
is a square of side [0, scale] (default: [0, 1])
Changing `center` shifts the layout by that amount.

For the other layout routines, the extent is
[center - scale, center + scale] (default: [-1, 1]).

Warning: Most layout routines have only been tested in 2-dimensions.

"""
from __future__ import division
import collections
import networkx as nx

__all__ = ['circular_layout',
           'kamada_kawai_layout',
           'random_layout',
           'rescale_layout',
           'shell_layout',
           'spring_layout',
           'spectral_layout',
           'fruchterman_reingold_layout']


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

    if not isinstance(G, nx.Graph):
        empty_graph = nx.Graph()
        empty_graph.add_nodes_from(G)
        G = empty_graph

    if center is None:
        center = np.zeros(dim)
    else:
        center = np.asarray(center)

    if len(center) != dim:
        msg = "length of center coordinates must match dimension of layout"
        raise ValueError(msg)

    return G, center


[docs]def random_layout(G, center=None, dim=2): """Position nodes uniformly at random in the unit square. For every node, a position is generated by choosing each of dim coordinates uniformly at random on the interval [0.0, 1.0). NumPy (http://scipy.org) is required for this function. Parameters ---------- G : NetworkX graph or list of nodes A position will be assigned to every node in G. center : array-like or None Coordinate pair around which to center the layout. dim : int Dimension of layout. Returns ------- pos : dict A dictionary of positions keyed by node Examples -------- >>> G = nx.lollipop_graph(4, 3) >>> pos = nx.random_layout(G) """ import numpy as np G, center = _process_params(G, center, dim) shape = (len(G), dim) pos = np.random.random(shape) + center pos = pos.astype(np.float32) pos = dict(zip(G, pos)) return pos
[docs]def circular_layout(G, scale=1, center=None, dim=2): # dim=2 only """Position nodes on a circle. Parameters ---------- G : NetworkX graph or list of nodes 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. Returns ------- pos : dict A dictionary of positions keyed by node Examples -------- >>> G = nx.path_graph(4) >>> pos = nx.circular_layout(G) Notes ----- This algorithm currently only works in two dimensions and does not try to minimize edge crossings. """ import numpy as np G, center = _process_params(G, center, dim) 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
[docs]def shell_layout(G, nlist=None, scale=1, center=None, dim=2): """Position nodes in concentric circles. Parameters ---------- G : NetworkX graph or list of nodes A position will be assigned to every node in G. nlist : list of lists List of node lists for each shell. 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. Returns ------- pos : dict A dictionary of positions keyed by node Examples -------- >>> G = nx.path_graph(4) >>> shells = [[0], [1, 2, 3]] >>> pos = nx.shell_layout(G, shells) Notes ----- This algorithm currently only works in two dimensions and does not try to minimize edge crossings. """ import numpy as np G, center = _process_params(G, center, dim) if len(G) == 0: return {} if len(G) == 1: return {nx.utils.arbitrary_element(G): center} if nlist is None: # draw the whole graph in one shell nlist = [list(G)] if len(nlist[0]) == 1: # single node at center radius = 0.0 else: # else start at r=1 radius = 1.0 npos = {} for nodes in nlist: # Discard the extra angle since it matches 0 radians. theta = np.linspace(0, 1, len(nodes) + 1)[:-1] * 2 * np.pi theta = theta.astype(np.float32) pos = np.column_stack([np.cos(theta), np.sin(theta)]) pos = rescale_layout(pos, scale=scale * radius / len(nlist)) + center npos.update(zip(nodes, pos)) radius += 1.0 return npos
def fruchterman_reingold_layout(G, k=None, pos=None, fixed=None, iterations=50, weight='weight', scale=1, center=None, dim=2): """Position nodes using Fruchterman-Reingold force-directed algorithm. 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. 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 : number (default: 1) Scale factor for positions. Not used unless `fixed is None`. center : array-like or None Coordinate pair around which to center the layout. Not used unless `fixed is None`. dim : int Dimension of layout. Returns ------- pos : dict A dictionary of positions keyed by node Examples -------- >>> G = nx.path_graph(4) >>> pos = nx.spring_layout(G) # The same using longer but equivalent function name >>> pos = nx.fruchterman_reingold_layout(G) """ import numpy as np G, center = _process_params(G, center, dim) if fixed is not None: nfixed = dict(zip(G, range(len(G)))) fixed = np.asarray([nfixed[v] for v in fixed]) if pos is not None: # Determine size of existing domain to adjust initial positions dom_size = max(coord for pos_tup in pos.values() for coord in pos_tup) if dom_size == 0: dom_size = 1 shape = (len(G), dim) pos_arr = np.random.random(shape) * dom_size + center for i, n in enumerate(G): if n in pos: pos_arr[i] = np.asarray(pos[n]) else: pos_arr = None if len(G) == 0: return {} if len(G) == 1: return {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_matrix(G, weight=weight, dtype='f') if k is None and fixed is not None: # We must adjust k by domain size for layouts not near 1x1 nnodes, _ = A.shape k = dom_size / np.sqrt(nnodes) pos = _sparse_fruchterman_reingold(A, k, pos_arr, fixed, iterations, dim) except: A = nx.to_numpy_matrix(G, weight=weight) if k is None and fixed is not None: # We must adjust k by domain size for layouts not near 1x1 nnodes, _ = A.shape k = dom_size / np.sqrt(nnodes) pos = _fruchterman_reingold(A, k, pos_arr, fixed, iterations, dim) if fixed is None: pos = rescale_layout(pos, scale=scale) + center pos = dict(zip(G, pos)) return pos spring_layout = fruchterman_reingold_layout def _fruchterman_reingold(A, k=None, pos=None, fixed=None, iterations=50, dim=2): # Position nodes in adjacency matrix A using Fruchterman-Reingold # Entry point for NetworkX graph is fruchterman_reingold_layout() try: import numpy as np except ImportError: msg = "_fruchterman_reingold() requires numpy: http://scipy.org/ " raise ImportError(msg) try: nnodes, _ = A.shape except AttributeError: msg = "fruchterman_reingold() takes an adjacency matrix as input" raise nx.NetworkXError(msg) # make sure we have an array instead of a matrix A = np.asarray(A) if pos is None: # random initial positions pos = np.asarray(np.random.random((nnodes, dim)), dtype=A.dtype) else: # make sure positions are of same type as matrix pos = pos.astype(A.dtype) # optimal distance between nodes if k is None: k = np.sqrt(1.0/nnodes) # the initial "temperature" is about .1 of domain area (=1x1) # this is the largest step allowed in the dynamics. # We need to calculate this in case our fixed positions force our domain # to be much bigger than 1x1 t = max(max(pos.T[0]) - min(pos.T[0]), max(pos.T[1]) - min(pos.T[1]))*0.1 # simple cooling scheme. # linearly step down by dt on each iteration so last iteration is size dt. dt = t/float(iterations+1) delta = np.zeros((pos.shape[0], pos.shape[0], pos.shape[1]), dtype=A.dtype) # the inscrutable (but fast) version # this is still O(V^2) # could use multilevel methods to speed this up significantly for iteration in range(iterations): # matrix of difference between points 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 return pos def _sparse_fruchterman_reingold(A, k=None, pos=None, fixed=None, iterations=50, dim=2): # Position nodes in adjacency matrix A using Fruchterman-Reingold # Entry point for NetworkX graph is fruchterman_reingold_layout() # Sparse version try: import numpy as np except ImportError: m = "_sparse_fruchterman_reingold() requires numpy: http://scipy.org/" raise ImportError(m) try: nnodes, _ = A.shape except AttributeError: msg = "fruchterman_reingold() takes an adjacency matrix as input" raise nx.NetworkXError(msg) try: from scipy.sparse import spdiags, coo_matrix except ImportError: msg = "_sparse_fruchterman_reingold() scipy numpy: http://scipy.org/ " raise ImportError(msg) # make sure we have a LIst of Lists representation try: A = A.tolil() except: A = (coo_matrix(A)).tolil() if pos is None: # random initial positions pos = np.asarray(np.random.random((nnodes, dim)), dtype=A.dtype) else: # make sure positions are of same type as matrix pos = pos.astype(A.dtype) # no fixed nodes if fixed is None: fixed = [] # optimal distance between nodes if k is None: k = np.sqrt(1.0/nnodes) # the initial "temperature" is about .1 of domain area (=1x1) # this is the largest step allowed in the dynamics. t = 0.1 # simple cooling scheme. # linearly step down by dt on each iteration so last iteration is size dt. dt = t / float(iterations+1) displacement = np.zeros((dim, nnodes)) for iteration in range(iterations): displacement *= 0 # loop over rows for i in range(A.shape[0]): if i in fixed: continue # difference between this row's node position and all others delta = (pos[i] - pos).T # distance between points distance = np.sqrt((delta**2).sum(axis=0)) # enforce minimum distance of 0.01 distance = np.where(distance < 0.01, 0.01, distance) # the adjacency matrix row Ai = np.asarray(A.getrowview(i).toarray()) # displacement "force" displacement[:, i] +=\ (delta * (k * k / distance**2 - Ai * distance / k)).sum(axis=1) # update positions length = np.sqrt((displacement**2).sum(axis=0)) length = np.where(length < 0.01, 0.1, length) pos += (displacement * t / length).T # cool temperature t -= dt return pos 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 : float (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(). 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) """ try: import numpy as np except ImportError: msg = 'Kamada-Kawai layout requires numpy: http://scipy.org' raise ImportError(msg) G, center = _process_params(G, center, dim) nNodes = len(G) 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: pos = circular_layout(G, dim=dim) 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 try: from scipy.optimize import minimize except ImportError: msg = 'Kamada-Kawai layout requires scipy: http://scipy.org' raise ImportError(msg) meanwt = 1e-3 costargs = (np, 1 / (dist_mtx + np.eye(dist_mtx.shape[0]) * 1e-3), meanwt, dim) optresult = 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())
[docs]def spectral_layout(G, weight='weight', scale=1, center=None, dim=2): """Position nodes using the eigenvectors of the graph Laplacian. Parameters ---------- G : NetworkX graph or list of nodes 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_matrix(G, weight=weight, dtype='d') # Symmetrize directed graphs if G.is_directed(): A = A + np.transpose(A) pos = _sparse_spectral(A, dim) except (ImportError, ValueError): # Dense matrix A = nx.to_numpy_matrix(G, weight=weight) # Symmetrize directed graphs if G.is_directed(): A = A + np.transpose(A) pos = _spectral(A, dim) pos = rescale_layout(pos, scale) + center pos = dict(zip(G, pos)) return pos
def _spectral(A, dim=2): # Input adjacency matrix A # Uses dense eigenvalue solver from numpy try: import numpy as np except ImportError: msg = "spectral_layout() requires numpy: http://scipy.org/ " raise ImportError(msg) try: nnodes, _ = A.shape except AttributeError: msg = "spectral() takes an adjacency matrix as input" raise nx.NetworkXError(msg) # form Laplacian matrix # make sure we have an array instead of a matrix A = np.asarray(A) I = np.identity(nnodes, dtype=A.dtype) D = I * np.sum(A, axis=1) # diagonal of degrees L = D - A eigenvalues, eigenvectors = np.linalg.eig(L) # sort and keep smallest nonzero index = np.argsort(eigenvalues)[1:dim + 1] # 0 index is zero eigenvalue return np.real(eigenvectors[:, index]) def _sparse_spectral(A, dim=2): # Input adjacency matrix A # Uses sparse eigenvalue solver from scipy # Could use multilevel methods here, see Koren "On spectral graph drawing" try: import numpy as np from scipy.sparse import spdiags from scipy.sparse.linalg.eigen import eigsh except ImportError: msg = "_sparse_spectral() requires scipy & numpy: http://scipy.org/ " raise ImportError(msg) try: nnodes, _ = A.shape except AttributeError: msg = "sparse_spectral() takes an adjacency matrix as input" raise nx.NetworkXError(msg) # form Laplacian matrix data = np.asarray(A.sum(axis=1).T) D = spdiags(data, 0, nnodes, nnodes) L = D - A k = dim + 1 # number of Lanczos vectors for ARPACK solver.What is the right scaling? ncv = max(2 * k + 1, int(np.sqrt(nnodes))) # return smallest k eigenvalues and eigenvectors eigenvalues, eigenvectors = eigsh(L, k, which='SM', ncv=ncv) index = np.argsort(eigenvalues)[1:k] # 0 index is zero eigenvalue return np.real(eigenvectors[:, index])
[docs]def rescale_layout(pos, scale=1): """Return scaled position array to (-scale, scale) in all axes. The function acts on NumPy arrays which hold position information. Each position is one row of the array. The dimension of the space equals the number of columns. Each coordinate in one column. To rescale, the mean (center) is subtracted from each axis separately. Then all values are scaled so that the largest magnitude value from all axes equals `scale` (thus, the aspect ratio is preserved). The resulting NumPy Array is returned (order of rows unchanged). Parameters ---------- pos : numpy array positions to be scaled. Each row is a position. scale : number (default: 1) The size of the resulting extent in all directions. Returns ------- pos : numpy array scaled positions. Each row is a position. """ # Find max length over all dimensions lim = 0 # max coordinate for all axes for i in range(pos.shape[1]): pos[:, i] -= pos[:, i].mean() lim = max(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
# 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")