venv/lib/python3.12/site-packages/networkx/drawing/layout.py

"""
******
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",
    "forceatlas2_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",
    "bfs_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()
        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, store_pos_as=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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.lollipop_graph(4, 3)
    >>> pos = nx.random_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.random_layout(G, seed=42, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([0.37454012, 0.9507143 ], dtype=float32),
     1: array([0.7319939, 0.5986585], dtype=float32),
     2: array([0.15601864, 0.15599452], dtype=float32),
     3: array([0.05808361, 0.8661761 ], dtype=float32),
     4: array([0.601115 , 0.7080726], dtype=float32),
     5: array([0.02058449, 0.96990985], dtype=float32),
     6: array([0.83244264, 0.21233912], dtype=float32)}
    """
    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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)
    return pos


def circular_layout(G, scale=1, center=None, dim=2, store_pos_as=None):
    # 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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.circular_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.circular_layout(G, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([9.99999986e-01, 2.18556937e-08]),
     1: array([-3.57647606e-08,  1.00000000e+00]),
     2: array([-9.9999997e-01, -6.5567081e-08]),
     3: array([ 1.98715071e-08, -9.99999956e-01])}


    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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos


def shell_layout(
    G, nlist=None, rotate=None, scale=1, center=None, dim=2, store_pos_as=None
):
    """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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> shells = [[0], [1, 2, 3]]
    >>> pos = nx.shell_layout(G, shells)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.shell_layout(G, shells, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([0., 0.]),
     1: array([-5.00000000e-01, -4.37113883e-08]),
     2: array([ 0.24999996, -0.43301272]),
     3: array([0.24999981, 0.43301281])}

    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

    if store_pos_as is not None:
        nx.set_node_attributes(G, npos, store_pos_as)
    return npos


def bipartite_layout(
    G,
    nodes=None,
    align="vertical",
    scale=1,
    center=None,
    aspect_ratio=4 / 3,
    store_pos_as=None,
):
    """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 : collection of nodes
        Nodes in one node set of the graph. This set will be placed on
        left or top. If `None` (the default), a node set is chosen arbitrarily
        if the graph if bipartite.

    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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Raises
    ------
    NetworkXError
        If ``nodes=None`` and `G` is not bipartite.

    Examples
    --------
    >>> G = nx.complete_bipartite_graph(3, 3)
    >>> pos = nx.bipartite_layout(G)

    The ordering of the layout (i.e. which nodes appear on the left/top) can
    be specified with the `nodes` parameter:

    >>> top, bottom = nx.bipartite.sets(G)
    >>> pos = nx.bipartite_layout(G, nodes=bottom)  # "bottom" set appears on the left

    `store_pos_as` can be used to store the node positions for the computed layout
    directly on the nodes:

    >>> _ = nx.bipartite_layout(G, nodes=bottom, store_pos_as="pos")
    >>> from pprint import pprint
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([ 1.  , -0.75]),
     1: array([1., 0.]),
     2: array([1.  , 0.75]),
     3: array([-1.  , -0.75]),
     4: array([-1.,  0.]),
     5: array([-1.  ,  0.75])}


    The ``bipartite_layout`` function can be used with non-bipartite graphs
    by explicitly specifying how the layout should be partitioned with `nodes`:

    >>> G = nx.complete_graph(5)  # Non-bipartite
    >>> pos = nx.bipartite_layout(G, nodes={0, 1, 2})

    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)

    if nodes is None:
        top, bottom = nx.bipartite.sets(G)
        nodes = list(G)
    else:
        top = set(nodes)
        bottom = set(G) - top
        # Preserves backward-compatible node ordering in returned pos dict
        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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos


@np_random_state("seed")
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,
    store_pos_as=None,
    *,
    method="auto",
    gravity=1.0,
):
    """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)
        Used only for the initial positions in the algorithm.
        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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

    method : str  optional (default='auto')
        The method to compute the layout.
        If 'force', the force-directed Fruchterman-Reingold algorithm [1]_ is used.
        If 'energy', the energy-based optimization algorithm [2]_ is used with absolute
        values of edge weights and gravitational forces acting on each connected component.
        If 'auto', we use 'force' if ``len(G) < 500`` and 'energy' otherwise.

    gravity: float optional (default=1.0)
        Used only for the method='energy'.
        The positive coefficient of gravitational forces per connected component.

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.spring_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.spring_layout(G, seed=123, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([-0.61495802, -1.        ]),
     1: array([-0.21789544, -0.35432583]),
     2: array([0.21847843, 0.35527369]),
     3: array([0.61437502, 0.99905215])}


    # The same using longer but equivalent function name
    >>> pos = nx.fruchterman_reingold_layout(G)

    References
    ----------
    .. [1] Fruchterman, Thomas MJ, and Edward M. Reingold.
           "Graph drawing by force-directed placement."
           Software: Practice and experience 21, no. 11 (1991): 1129-1164.
           http://dx.doi.org/10.1002/spe.4380211102
    .. [2] Hamaguchi, Hiroki, Naoki Marumo, and Akiko Takeda.
           "Initial Placement for Fruchterman--Reingold Force Model With Coordinate Newton Direction."
           arXiv preprint arXiv:2412.20317 (2024).
           https://arxiv.org/abs/2412.20317
    """
    import numpy as np

    if method not in ("auto", "force", "energy"):
        raise ValueError("the method must be either auto, force, or energy.")
    if method == "auto":
        method = "force" if len(G) < 500 else "energy"

    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:
        pos = {nx.utils.arbitrary_element(G.nodes()): center}
        if store_pos_as is not None:
            nx.set_node_attributes(G, pos, store_pos_as)
        return pos

    # Sparse matrix
    if len(G) >= 500 or method == "energy":
        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, method, gravity
        )
    else:
        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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    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 / (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)
        # Threshold the minimum length prior to position scaling
        # See gh-8113 for detailed discussion of the threshold
        length = np.clip(length, a_min=0.01, a_max=None)
        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,
    method="energy",
    gravity=1.0,
):
    # 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

    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)

    # no fixed nodes
    if fixed is None:
        fixed = []

    # optimal distance between nodes
    if k is None:
        k = np.sqrt(1.0 / nnodes)

    if method == "energy":
        return _energy_fruchterman_reingold(
            A, nnodes, k, pos, fixed, iterations, threshold, dim, gravity
        )

    # make sure we have a LIst of Lists representation
    try:
        A = A.tolil()
    except AttributeError:
        A = (sp.sparse.coo_array(A)).tolil()

    # 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 / (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.clip(distance, a_min=0.01, a_max=None)
            # 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))
        # Threshold the minimum length prior to position scaling
        # See gh-8113 for detailed discussion of the threshold
        length = np.clip(length, a_min=0.01, a_max=None)
        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 _energy_fruchterman_reingold(
    A, nnodes, k, pos, fixed, iterations, threshold, dim, gravity
):
    # Entry point for NetworkX graph is fruchterman_reingold_layout()
    # energy-based version
    import numpy as np
    import scipy as sp

    if gravity <= 0:
        raise ValueError(f"the gravity must be positive.")

    # make sure we have a Compressed Sparse Row format
    try:
        A = A.tocsr()
    except AttributeError:
        A = sp.sparse.csr_array(A)

    # Take absolute values of edge weights and symmetrize it
    A = np.abs(A)
    A = (A + A.T) / 2

    n_components, labels = sp.sparse.csgraph.connected_components(A, directed=False)
    bincount = np.bincount(labels)
    batchsize = 500

    def _cost_FR(x):
        pos = x.reshape((nnodes, dim))
        grad = np.zeros((nnodes, dim))
        cost = 0.0
        for l in range(0, nnodes, batchsize):
            r = min(l + batchsize, nnodes)
            # difference between selected node positions and all others
            delta = pos[l:r, np.newaxis, :] - pos[np.newaxis, :, :]
            # distance between points with a minimum distance of 1e-5
            distance2 = np.sum(delta * delta, axis=2)
            distance2 = np.maximum(distance2, 1e-10)
            distance = np.sqrt(distance2)
            # temporary variable for calculation
            Ad = A[l:r] * distance
            # attractive forces and repulsive forces
            grad[l:r] = 2 * np.einsum("ij,ijk->ik", Ad / k - k**2 / distance2, delta)
            # integrated attractive forces
            cost += np.sum(Ad * distance2) / (3 * k)
            # integrated repulsive forces
            cost -= k**2 * np.sum(np.log(distance))
        # gravitational force from the centroids of connected components to (0.5, ..., 0.5)^T
        centers = np.zeros((n_components, dim))
        np.add.at(centers, labels, pos)
        delta0 = centers / bincount[:, np.newaxis] - 0.5
        grad += gravity * delta0[labels]
        cost += gravity * 0.5 * np.sum(bincount * np.linalg.norm(delta0, axis=1) ** 2)
        # fix positions of fixed nodes
        grad[fixed] = 0.0
        return cost, grad.ravel()

    # Optimization of the energy function by L-BFGS algorithm
    options = {"maxiter": iterations, "gtol": threshold}
    return sp.optimize.minimize(
        _cost_FR, pos.ravel(), method="L-BFGS-B", jac=True, options=options
    ).x.reshape((nnodes, dim))


def kamada_kawai_layout(
    G,
    dist=None,
    pos=None,
    weight="weight",
    scale=1,
    center=None,
    dim=2,
    store_pos_as=None,
):
    """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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.kamada_kawai_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.kamada_kawai_layout(G, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([0.99996577, 0.99366857]),
     1: array([0.32913544, 0.33543827]),
     2: array([-0.33544334, -0.32910684]),
     3: array([-0.99365787, -1.        ])}
    """
    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 = _kamada_kawai_solve(dist_mtx, pos_arr, dim)

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

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return 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

    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, store_pos_as=None):
    """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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.spectral_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.spectral_layout(G, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([-1.        ,  0.76536686]),
     1: array([-0.41421356, -0.76536686]),
     2: array([ 0.41421356, -0.76536686]),
     3: array([1.        , 0.76536686])}


    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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    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

    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
    D = sp.sparse.dia_array((A.sum(axis=1), 0), shape=(nnodes, nnodes)).tocsr()
    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, store_pos_as=None):
    """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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.planar_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.planar_layout(G, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([-0.77777778, -0.33333333]),
     1: array([ 1.        , -0.33333333]),
     2: array([0.11111111, 0.55555556]),
     3: array([-0.33333333,  0.11111111])}
    """
    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.vstack([pos[x] for x in node_list])
    pos = pos.astype(np.float64)
    pos = rescale_layout(pos, scale=scale) + center
    pos = dict(zip(node_list, pos))
    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)
    return pos


def spiral_layout(
    G,
    scale=1,
    center=None,
    dim=2,
    resolution=0.35,
    equidistant=False,
    store_pos_as=None,
):
    """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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.spiral_layout(G)
    >>> nx.draw(G, pos=pos)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.spiral_layout(G, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([-0.64153279, -0.68555087]),
     1: array([-0.03307913, -0.46344795]),
     2: array([0.34927952, 0.14899882]),
     3: array([0.32533239, 1.        ])}

    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:
        pos = {nx.utils.arbitrary_element(G): center}
        if store_pos_as is not None:
            nx.set_node_attributes(G, pos, store_pos_as)
        return pos

    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))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos


def multipartite_layout(
    G, subset_key="subset", align="vertical", scale=1, center=None, store_pos_as=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 or dict (default='subset')
        If a string, the key of node data in G that holds the node subset.
        If a dict, keyed by layer number to the nodes in that 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.

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> G = nx.complete_multipartite_graph(28, 16, 10)
    >>> pos = nx.multipartite_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> G = nx.complete_multipartite_graph(28, 16, 10)
    >>> _ = nx.multipartite_layout(G, store_pos_as="pos")

    or use a dict to provide the layers of the layout

    >>> G = nx.Graph([(0, 1), (1, 2), (1, 3), (3, 4)])
    >>> layers = {"a": [0], "b": [1], "c": [2, 3], "d": [4]}
    >>> pos = nx.multipartite_layout(G, subset_key=layers)

    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 {}

    try:
        # check if subset_key is dict-like
        if len(G) != sum(len(nodes) for nodes in subset_key.values()):
            raise nx.NetworkXError(
                "all nodes must be in one subset of `subset_key` dict"
            )
    except AttributeError:
        # subset_key is not a dict, hence a string
        node_to_subset = nx.get_node_attributes(G, subset_key)
        if len(node_to_subset) != len(G):
            raise nx.NetworkXError(
                f"all nodes need a subset_key attribute: {subset_key}"
            )
        subset_key = nx.utils.groups(node_to_subset)

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

    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 = pos[:, ::-1]  # swap x and y coords
    pos = dict(zip(nodes, pos))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos


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

    The attractive and repulsive forces (arf) layout [1] 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 symmetries 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 of unconnected sub clusters.
    etol : float
        Gradient 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.
    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.
    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> G = nx.grid_graph((5, 5))
    >>> pos = nx.arf_layout(G)
    >>> # suppress the returned dict and store on the graph directly
    >>> G = nx.grid_graph((5, 5))
    >>> _ = nx.arf_layout(G, store_pos_as="pos")

    References
    ----------
    .. [1] "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
    """
    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, seed=seed)
    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 [1]
    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

    pos = dict(zip(G.nodes(), p))

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos


@np_random_state("seed")
@nx._dispatchable(edge_attrs="weight", mutates_input={"store_pos_as": 15})
def forceatlas2_layout(
    G,
    pos=None,
    *,
    max_iter=100,
    jitter_tolerance=1.0,
    scaling_ratio=2.0,
    gravity=1.0,
    distributed_action=False,
    strong_gravity=False,
    node_mass=None,
    node_size=None,
    weight=None,
    linlog=False,
    seed=None,
    dim=2,
    store_pos_as=None,
):
    """Position nodes using the ForceAtlas2 force-directed layout algorithm.

    This function applies the ForceAtlas2 layout algorithm [1]_ to a NetworkX graph,
    positioning the nodes in a way that visually represents the structure of the graph.
    The algorithm uses physical simulation to minimize the energy of the system,
    resulting in a more readable layout.

    Parameters
    ----------
    G : nx.Graph
        A NetworkX graph to be laid out.
    pos : dict or None, optional
        Initial positions of the nodes. If None, random initial positions are used.
    max_iter : int (default: 100)
        Number of iterations for the layout optimization.
    jitter_tolerance : float (default: 1.0)
        Controls the tolerance for adjusting the speed of layout generation.
    scaling_ratio : float (default: 2.0)
        Determines the scaling of attraction and repulsion forces.
    gravity : float (default: 1.0)
        Determines the amount of attraction on nodes to the center. Prevents islands
        (i.e. weakly connected or disconnected parts of the graph)
        from drifting away.
    distributed_action : bool (default: False)
        Distributes the attraction force evenly among nodes.
    strong_gravity : bool (default: False)
        Applies a strong gravitational pull towards the center.
    node_mass : dict or None, optional
        Maps nodes to their masses, influencing the attraction to other nodes.
    node_size : dict or None, optional
        Maps nodes to their sizes, preventing crowding by creating a halo effect.
    weight : string or None, optional (default: None)
        The edge attribute that holds the numerical value used for
        the edge weight. If None, then all edge weights are 1.
    linlog : bool (default: False)
        Uses logarithmic attraction instead of linear.
    seed : int, RandomState instance or None  optional (default=None)
        Used only for the initial positions in the algorithm.
        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.
    dim : int (default: 2)
        Sets the dimensions for the layout. Ignored if `pos` is provided.
    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.florentine_families_graph()
    >>> pos = nx.forceatlas2_layout(G)
    >>> nx.draw(G, pos=pos)
    >>> # suppress the returned dict and store on the graph directly
    >>> pos = nx.forceatlas2_layout(G, store_pos_as="pos")
    >>> _ = nx.forceatlas2_layout(G, store_pos_as="pos")

    References
    ----------
    .. [1] Jacomy, M., Venturini, T., Heymann, S., & Bastian, M. (2014).
           ForceAtlas2, a continuous graph layout algorithm for handy network
           visualization designed for the Gephi software. PloS one, 9(6), e98679.
           https://doi.org/10.1371/journal.pone.0098679
    """
    import numpy as np

    if len(G) == 0:
        return {}
    # parse optional pos positions
    if pos is None:
        pos = nx.random_layout(G, dim=dim, seed=seed)
        pos_arr = np.array(list(pos.values()))
    elif len(pos) == len(G):
        pos_arr = np.array([pos[node].copy() for node in G])
    else:
        # set random node pos within the initial pos values
        pos_init = np.array(list(pos.values()))
        max_pos = pos_init.max(axis=0)
        min_pos = pos_init.min(axis=0)
        dim = max_pos.size
        pos_arr = min_pos + seed.rand(len(G), dim) * (max_pos - min_pos)
        for idx, node in enumerate(G):
            if node in pos:
                pos_arr[idx] = pos[node].copy()

    mass = np.zeros(len(G))
    size = np.zeros(len(G))

    # Only adjust for size when the users specifies size other than default (1)
    adjust_sizes = False
    if node_size is None:
        node_size = {}
    else:
        adjust_sizes = True

    if node_mass is None:
        node_mass = {}

    for idx, node in enumerate(G):
        mass[idx] = node_mass.get(node, G.degree(node) + 1)
        size[idx] = node_size.get(node, 1)

    n = len(G)
    gravities = np.zeros((n, dim))
    attraction = np.zeros((n, dim))
    repulsion = np.zeros((n, dim))
    A = nx.to_numpy_array(G, weight=weight)

    def estimate_factor(n, swing, traction, speed, speed_efficiency, jitter_tolerance):
        """Computes the scaling factor for the force in the ForceAtlas2 layout algorithm.

        This   helper  function   adjusts   the  speed   and
        efficiency  of the  layout generation  based on  the
        current state of  the system, such as  the number of
        nodes, current swing, and traction forces.

        Parameters
        ----------
        n : int
            Number of nodes in the graph.
        swing : float
            The current swing, representing the oscillation of the nodes.
        traction : float
            The current traction force, representing the attraction between nodes.
        speed : float
            The current speed of the layout generation.
        speed_efficiency : float
            The efficiency of the current speed, influencing how fast the layout converges.
        jitter_tolerance : float
            The tolerance for jitter, affecting how much speed adjustment is allowed.

        Returns
        -------
        tuple
            A tuple containing the updated speed and speed efficiency.

        Notes
        -----
        This function is a part of the ForceAtlas2 layout algorithm and is used to dynamically adjust the
        layout parameters to achieve an optimal and stable visualization.

        """
        import numpy as np

        # estimate jitter
        opt_jitter = 0.05 * np.sqrt(n)
        min_jitter = np.sqrt(opt_jitter)
        max_jitter = 10
        min_speed_efficiency = 0.05

        other = min(max_jitter, opt_jitter * traction / n**2)
        jitter = jitter_tolerance * max(min_jitter, other)

        if swing / traction > 2.0:
            if speed_efficiency > min_speed_efficiency:
                speed_efficiency *= 0.5
            jitter = max(jitter, jitter_tolerance)
        if swing == 0:
            target_speed = np.inf
        else:
            target_speed = jitter * speed_efficiency * traction / swing

        if swing > jitter * traction:
            if speed_efficiency > min_speed_efficiency:
                speed_efficiency *= 0.7
        elif speed < 1000:
            speed_efficiency *= 1.3

        max_rise = 0.5
        speed = speed + min(target_speed - speed, max_rise * speed)
        return speed, speed_efficiency

    speed = 1
    speed_efficiency = 1
    swing = 1
    traction = 1
    for _ in range(max_iter):
        # compute pairwise difference
        diff = pos_arr[:, None] - pos_arr[None]
        # compute pairwise distance
        distance = np.linalg.norm(diff, axis=-1)

        # linear attraction
        if linlog:
            attraction = -np.log(1 + distance) / distance
            np.fill_diagonal(attraction, 0)
            attraction = np.einsum("ij, ij -> ij", attraction, A)
            attraction = np.einsum("ijk, ij -> ik", diff, attraction)

        else:
            attraction = -np.einsum("ijk, ij -> ik", diff, A)

        if distributed_action:
            attraction /= mass[:, None]

        # repulsion
        tmp = mass[:, None] @ mass[None]
        if adjust_sizes:
            distance += -size[:, None] - size[None]

        d2 = distance**2
        # remove self-interaction
        np.fill_diagonal(tmp, 0)
        np.fill_diagonal(d2, 1)
        factor = (tmp / d2) * scaling_ratio
        repulsion = np.einsum("ijk, ij -> ik", diff, factor)

        # gravity
        pos_centered = pos_arr - np.mean(pos_arr, axis=0)
        if strong_gravity:
            gravities = -gravity * mass[:, None] * pos_centered
        else:
            # hide warnings for divide by zero. Then change nan to 0
            with np.errstate(divide="ignore", invalid="ignore"):
                unit_vec = pos_centered / np.linalg.norm(pos_centered, axis=-1)[:, None]
            unit_vec = np.nan_to_num(unit_vec, nan=0)
            gravities = -gravity * mass[:, None] * unit_vec

        # total forces
        update = attraction + repulsion + gravities

        # compute total swing and traction
        swing += (mass * np.linalg.norm(pos_arr - update, axis=-1)).sum()
        traction += (0.5 * mass * np.linalg.norm(pos_arr + update, axis=-1)).sum()

        speed, speed_efficiency = estimate_factor(
            n,
            swing,
            traction,
            speed,
            speed_efficiency,
            jitter_tolerance,
        )

        # update pos
        if adjust_sizes:
            df = np.linalg.norm(update, axis=-1)
            swinging = mass * df
            factor = 0.1 * speed / (1 + np.sqrt(speed * swinging))
            factor = np.minimum(factor * df, 10.0 * np.ones(df.shape)) / df
        else:
            swinging = mass * np.linalg.norm(update, axis=-1)
            factor = speed / (1 + np.sqrt(speed * swinging))

        factored_update = update * factor[:, None]
        pos_arr += factored_update
        if abs(factored_update).sum() < 1e-10:
            break

    pos = dict(zip(G, pos_arr))
    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    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.

    attribute : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute named `attribute` which can be accessed with
        `G.nodes[...][attribute]`. The function still returns the dictionary.

    Returns
    -------
    pos : numpy array
        scaled positions. Each row is a position.

    See Also
    --------
    rescale_layout_dict
    """
    import numpy as np

    # Find max length over all dimensions
    pos -= pos.mean(axis=0)
    lim = np.abs(pos).max()  # max coordinate for all axes
    # rescale to (-scale, scale) in all directions, preserves aspect
    if lim > 0:
        pos *= 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))


def bfs_layout(G, start, *, align="vertical", scale=1, center=None, store_pos_as=None):
    """Position nodes according to breadth-first search algorithm.

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

    start : node in `G`
        Starting node for bfs

    align : string (default='vertical')
        The alignment of nodes within a layer, either `"vertical"` or
        `"horizontal"`.

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

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

    store_pos_as : str, default None
        If non-None, the position of each node will be stored on the graph as
        an attribute with this string as its name, which can be accessed with
        ``G.nodes[...][store_pos_as]``. The function still returns the dictionary.

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

    Examples
    --------
    >>> from pprint import pprint
    >>> G = nx.path_graph(4)
    >>> pos = nx.bfs_layout(G, 0)
    >>> # suppress the returned dict and store on the graph directly
    >>> _ = nx.bfs_layout(G, 0, store_pos_as="pos")
    >>> pprint(nx.get_node_attributes(G, "pos"))
    {0: array([-1.,  0.]),
     1: array([-0.33333333,  0.        ]),
     2: array([0.33333333, 0.        ]),
     3: array([1., 0.])}



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

    """
    G, center = _process_params(G, center, 2)

    # Compute layers with BFS
    layers = dict(enumerate(nx.bfs_layers(G, start)))

    if len(G) != sum(len(nodes) for nodes in layers.values()):
        raise nx.NetworkXError(
            "bfs_layout didn't include all nodes. Perhaps use input graph:\n"
            "        G.subgraph(nx.node_connected_component(G, start))"
        )

    # Compute node positions with multipartite_layout
    pos = multipartite_layout(
        G, subset_key=layers, align=align, scale=scale, center=center
    )

    if store_pos_as is not None:
        nx.set_node_attributes(G, pos, store_pos_as)

    return pos