__init__.py 8.51 KB
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#! /usr/bin/env python
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# -*- coding: utf-8 -*-
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#
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# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2007-2010 Tiago de Paula Peixoto <tiago@forked.de>
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#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

"""
``graph_tool.spectral`` - Spectral properties
---------------------------------------------
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Summary
+++++++

.. autosummary::
   :nosignatures:

   adjacency
   laplacian
   incidence

Contents
++++++++
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"""

from .. core import _degree, _prop, Graph, _limit_args
from numpy import *
import scipy.sparse


__all__ = ["adjacency", "laplacian", "incidence"]

def adjacency(g, sparse=True, weight=None):
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    r"""Return the adjacency matrix of the graph.

    Parameters
    ----------
    g : Graph
        Graph to be used.
    sparse : bool (optional, default: True)
        Build a :mod:`~scipy.sparse` matrix.
    weight : PropertyMap (optional, default: True)
        Edge property map with the edge weights.

    Returns
    -------
    a : matrix
        The adjacency matrix.

    Notes
    -----
    The adjacency matrix is defined as

    .. math::

        a_{i,j} =
        \begin{cases}
            1 & \text{if } v_i \text{ is adjacent to } v_j, \\
            0 & \text{otherwise}
        \end{cases}

    In the case of weighted edges, the value 1 is replaced the weight of the
    respective edge.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (10,10))
    >>> m = gt.adjacency(g)
    >>> print m.todense()
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    [[ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]
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     [ 0.  0.  0. ...,  0.  0.  0.]
     ..., 
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     [ 0.  0.  0. ...,  0.  1.  1.]
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     [ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]]

    References
    ----------
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    .. [wikipedia-adjacency] http://en.wikipedia.org/wiki/Adjacency_matrix
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    """

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    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
        for i,v in enumerate(g.vertices()):
            index[v] = i
    else:
        index = g.vertex_index
    N = g.num_vertices()
    if sparse:
        m = scipy.sparse.lil_matrix((N,N))
    else:
        m = matrix(zeros((N,N)))
    for v in g.vertices():
        for e in v.out_edges():
            m[index[v],index[e.target()]] = 1 if weight == None else weight[e]
    if sparse:
        m = m.tocsr()
    return m

def _get_deg(v, deg, weight):
    if deg == "total":
        if weight == None:
            d = v.in_degree() + v.out_degree()
        else:
            d = sum(weight[e] for e in v.all_edges())
    elif deg == "in":
        if weight == None:
            d = v.in_degree()
        else:
            d = sum(weight[e] for e in v.in_edges())
    else:
        if weight == None:
            d = v.out_degree()
        else:
            d = sum(weight[e] for e in v.out_edges())
    return d

@_limit_args({"deg":["total", "in", "out"]})
def laplacian(g, deg="total", normalized=True, sparse=True, weight=None):
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    r"""Return the Laplacian matrix of the graph.

    Parameters
    ----------
    g : Graph
        Graph to be used.
    deg : str (optional, default: "total")
        Degree to be used, in case of a directed graph.
    normalized : bool (optional, default: True)
        Whether to compute the normalized Laplacian.
    sparse : bool (optional, default: True)
        Build a :mod:`~scipy.sparse` matrix.
    weight : PropertyMap (optional, default: True)
        Edge property map with the edge weights.

    Returns
    -------
    l : matrix
        The Laplacian matrix.

    Notes
    -----
    The Laplacian matrix is defined as

    .. math::

        \ell_{i,j} =
        \begin{cases}
        \Gamma(v_i) & \text{if } i = j \\
        -1          & \text{if } i \neq j \text{ and } v_i \text{ is adjacent to } v_j \\
        0           & \text{otherwise}.
        \end{cases}

    Where :math:`\Gamma(v_i)` is the degree of vertex :math:`v_i`. The
    normalized version is

    .. math::

        \ell_{i,j} =
        \begin{cases}
        1         & \text{ if } i = j \text{ and } \Gamma(v_i) \neq 0 \\
       -\frac{1}{\sqrt{\Gamma(v_i)\Gamma(v_j)}} & \text{ if } i \neq j \text{ and } v_i \text{ is adjacent to } v_j \\
        0         & \text{otherwise}.
        \end{cases}

    In the case of weighted edges, the value 1 is replaced the weight of the
    respective edge.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (10,10))
    >>> m = gt.laplacian(g)
    >>> print m.todense()
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    [[ 1.    0.    0.   ...,  0.    0.    0.  ]
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     [ 0.    1.    0.   ...,  0.    0.    0.  ]
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     [ 0.    0.    1.   ...,  0.    0.    0.  ]
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     ..., 
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     [ 0.    0.    0.   ...,  1.    0.05  0.05]
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     [ 0.    0.    0.   ...,  0.    1.    0.  ]
     [ 0.    0.    0.   ...,  0.    0.    1.  ]]

    References
    ----------
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    .. [wikipedia-laplacian] http://en.wikipedia.org/wiki/Laplacian_matrix
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    """

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    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
        for i,v in enumerate(g.vertices()):
            index[v] = i
    else:
        index = g.vertex_index
    N = g.num_vertices()
    if sparse:
        m = scipy.sparse.lil_matrix((N,N))
    else:
        m = matrix(zeros((N,N)))
    for v in g.vertices():
        d = _get_deg(v, deg, weight)
        if not normalized:
            m[index[v], index[v]] = d
        elif d > 0:
            m[index[v], index[v]] = 1
        for e in v.out_edges():
            if not normalized:
                m[index[v],index[e.target()]] = (-1 if weight == None
                                                 else -weight[e])
            else:
                val = (d*_get_deg(e.target(),deg,weight))**(-0.5)
                m[index[v],index[e.target()]] = val
    if sparse:
        m = m.tocsr()
    return m

def incidence(g, sparse=True):
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    r"""Return the incidence matrix of the graph.

    Parameters
    ----------
    g : Graph
        Graph to be used.
    sparse : bool (optional, default: True)
        Build a :mod:`~scipy.sparse` matrix.

    Returns
    -------
    a : matrix
        The adjacency matrix.

    Notes
    -----
    For undirected graphs, the incidence matrix is defined as

    .. math::

        b_{i,j} =
        \begin{cases}
            1 & \text{if vertex } v_i \text{and edge } e_j \text{ are incident}, \\
            0 & \text{otherwise}
        \end{cases}

    For directed graphs, the definition is

    .. math::

        b_{i,j} =
        \begin{cases}
            1 & \text{if edge } e_j \text{ enters vertex } v_i, \\
            -1 & \text{if edge } e_j \text{ leaves vertex } v_i, \\
            0 & \text{otherwise}
        \end{cases}

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (2,2))
    >>> m = gt.incidence(g)
    >>> print m.todense()
    [[ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]
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     [ 0.  0.  0. ...,  0.  0.  1.]
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     ..., 
     [ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]]

    References
    ----------
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    .. [wikipedia-incidence] http://en.wikipedia.org/wiki/Incidence_matrix
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    """

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    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
        for i,v in enumerate(g.vertices()):
            index[v] = i
    else:
        index = g.vertex_index

    eindex = g.new_edge_property("int64_t")
    for i, e in enumerate(g.edges()):
        eindex[e] = i

    N = g.num_vertices()
    E = g.num_edges()
    if sparse:
        m = scipy.sparse.lil_matrix((N,E))
    else:
        m = matrix(zeros((N,E)))
    for v in g.vertices():
        if g.is_directed():
            for e in v.out_edges():
                m[index[v],eindex[e]] += -1
            for e in v.in_edges():
                m[index[v],eindex[e]] += 1
        else:
            for e in v.out_edges():
                m[index[v],eindex[e]] += 1
    if sparse:
        m = m.tocsr()
    return m