__init__.py 8.93 KB
Newer Older
Tiago Peixoto's avatar
Tiago Peixoto committed
1
#! /usr/bin/env python
2
# -*- coding: utf-8 -*-
Tiago Peixoto's avatar
Tiago Peixoto committed
3
#
4
5
# graph_tool -- a general graph manipulation python module
#
Tiago Peixoto's avatar
Tiago Peixoto committed
6
# Copyright (C) 2006-2013 Tiago de Paula Peixoto <tiago@skewed.de>
Tiago Peixoto's avatar
Tiago Peixoto committed
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
#
# 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
---------------------------------------------
24
25
26
27
28
29
30
31
32
33
34
35
36

Summary
+++++++

.. autosummary::
   :nosignatures:

   adjacency
   laplacian
   incidence

Contents
++++++++
Tiago Peixoto's avatar
Tiago Peixoto committed
37
38
"""

39
40
from __future__ import division, absolute_import, print_function

41
from .. import _degree, _prop, Graph, _limit_args
Tiago Peixoto's avatar
Tiago Peixoto committed
42
43
44
45
46
47
from numpy import *
import scipy.sparse


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

48

Tiago Peixoto's avatar
Tiago Peixoto committed
49
def adjacency(g, sparse=True, weight=None):
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
    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.

81
82
83
    In the case of networks with parallel edges, the entries in the matrix
    become simply the edge multiplicities.

84
85
    Examples
    --------
86
87
88
89
90
    .. testsetup::

      gt.seed_rng(42)

    >>> g = gt.random_graph(100, lambda: (10, 10))
91
    >>> m = gt.adjacency(g)
92
    >>> print(m.todense())
Tiago Peixoto's avatar
Tiago Peixoto committed
93
    [[ 0.  0.  0. ...,  0.  0.  0.]
94
     [ 0.  0.  0. ...,  0.  0.  0.]
Tiago Peixoto's avatar
Tiago Peixoto committed
95
96
     [ 0.  0.  0. ...,  1.  0.  1.]
     ..., 
97
     [ 0.  0.  0. ...,  0.  0.  0.]
Tiago Peixoto's avatar
Tiago Peixoto committed
98
     [ 0.  0.  1. ...,  0.  0.  0.]
Tiago Peixoto's avatar
Tiago Peixoto committed
99
     [ 0.  0.  0. ...,  1.  0.  0.]]
100
101
102

    References
    ----------
103
    .. [wikipedia-adjacency] http://en.wikipedia.org/wiki/Adjacency_matrix
104
105
    """

Tiago Peixoto's avatar
Tiago Peixoto committed
106
107
    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
Tiago Peixoto's avatar
Tiago Peixoto committed
108
        for i, v in enumerate(g.vertices()):
Tiago Peixoto's avatar
Tiago Peixoto committed
109
110
111
112
113
            index[v] = i
    else:
        index = g.vertex_index
    N = g.num_vertices()
    if sparse:
Tiago Peixoto's avatar
Tiago Peixoto committed
114
        m = scipy.sparse.lil_matrix((N, N))
Tiago Peixoto's avatar
Tiago Peixoto committed
115
    else:
Tiago Peixoto's avatar
Tiago Peixoto committed
116
        m = matrix(zeros((N, N)))
Tiago Peixoto's avatar
Tiago Peixoto committed
117
118
    for v in g.vertices():
        for e in v.out_edges():
119
            m[index[v], index[e.target()]] += 1 if weight is None else weight[e]
Tiago Peixoto's avatar
Tiago Peixoto committed
120
121
122
123
    if sparse:
        m = m.tocsr()
    return m

Tiago Peixoto's avatar
Tiago Peixoto committed
124

Tiago Peixoto's avatar
Tiago Peixoto committed
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
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

Tiago Peixoto's avatar
Tiago Peixoto committed
143
144

@_limit_args({"deg": ["total", "in", "out"]})
Tiago Peixoto's avatar
Tiago Peixoto committed
145
def laplacian(g, deg="total", normalized=True, sparse=True, weight=None):
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
    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
    --------
196
197
198
199
    .. testsetup::

      gt.seed_rng(42)

200
201
    >>> g = gt.random_graph(100, lambda: (10,10))
    >>> m = gt.laplacian(g)
202
    >>> print(m.todense())
Tiago Peixoto's avatar
Tiago Peixoto committed
203
    [[ 1.    0.    0.   ...,  0.    0.    0.  ]
Tiago Peixoto's avatar
Tiago Peixoto committed
204
     [ 0.    1.    0.   ...,  0.    0.    0.  ]
Tiago Peixoto's avatar
Tiago Peixoto committed
205
206
207
208
     [ 0.    0.    1.   ...,  0.    0.    0.05]
     ..., 
     [ 0.    0.    0.05 ...,  1.    0.    0.05]
     [ 0.05  0.    0.   ...,  0.    1.    0.  ]
Tiago Peixoto's avatar
Tiago Peixoto committed
209
     [ 0.    0.05  0.   ...,  0.    0.    1.  ]]
210
211
212

    References
    ----------
213
    .. [wikipedia-laplacian] http://en.wikipedia.org/wiki/Laplacian_matrix
214
215
    """

Tiago Peixoto's avatar
Tiago Peixoto committed
216
217
    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
Tiago Peixoto's avatar
Tiago Peixoto committed
218
        for i, v in enumerate(g.vertices()):
Tiago Peixoto's avatar
Tiago Peixoto committed
219
220
221
222
223
            index[v] = i
    else:
        index = g.vertex_index
    N = g.num_vertices()
    if sparse:
Tiago Peixoto's avatar
Tiago Peixoto committed
224
        m = scipy.sparse.lil_matrix((N, N))
Tiago Peixoto's avatar
Tiago Peixoto committed
225
    else:
Tiago Peixoto's avatar
Tiago Peixoto committed
226
        m = matrix(zeros((N, N)))
Tiago Peixoto's avatar
Tiago Peixoto committed
227
228
229
230
    for v in g.vertices():
        d = _get_deg(v, deg, weight)
        for e in v.out_edges():
            if not normalized:
231
232
233
234
235
236
                if weight is None:
                    val = -1
                else:
                    val = -weight[e]
                # increment in case of parallel edges
                m[index[v], index[e.target()]] += val
Tiago Peixoto's avatar
Tiago Peixoto committed
237
            else:
238
239
240
241
242
243
244
                d2 = _get_deg(e.target(), deg, weight)
                if weight is None:
                    w = 1
                else:
                    w = weight[e]
                # increment in case of parallel edges
                m[index[v], index[e.target()]] += - w / sqrt(d * d2)
245
246
247
        if not normalized:
            m[index[v], index[v]] = d
        elif d > 0:
248
            m[index[v], index[v]] = 1 if d != 0 else 0
Tiago Peixoto's avatar
Tiago Peixoto committed
249
250
251
252
    if sparse:
        m = m.tocsr()
    return m

Tiago Peixoto's avatar
Tiago Peixoto committed
253

Tiago Peixoto's avatar
Tiago Peixoto committed
254
def incidence(g, sparse=True):
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
    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
    --------
294
295
296
297
    .. testsetup::

      gt.seed_rng(42)

298
299
    >>> g = gt.random_graph(100, lambda: (2,2))
    >>> m = gt.incidence(g)
300
    >>> print(m.todense())
301
302
    [[ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]
303
     [ 0.  0.  0. ...,  0.  0.  0.]
Tiago Peixoto's avatar
Tiago Peixoto committed
304
     ..., 
305
306
307
308
309
310
     [ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]
     [ 0.  0.  0. ...,  0.  0.  0.]]

    References
    ----------
311
    .. [wikipedia-incidence] http://en.wikipedia.org/wiki/Incidence_matrix
312
313
    """

Tiago Peixoto's avatar
Tiago Peixoto committed
314
315
    if g.get_vertex_filter()[0] != None:
        index = g.new_vertex_property("int64_t")
Tiago Peixoto's avatar
Tiago Peixoto committed
316
        for i, v in enumerate(g.vertices()):
Tiago Peixoto's avatar
Tiago Peixoto committed
317
318
319
320
321
322
323
324
325
326
327
            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:
Tiago Peixoto's avatar
Tiago Peixoto committed
328
        m = scipy.sparse.lil_matrix((N, E))
Tiago Peixoto's avatar
Tiago Peixoto committed
329
    else:
Tiago Peixoto's avatar
Tiago Peixoto committed
330
        m = matrix(zeros((N, E)))
Tiago Peixoto's avatar
Tiago Peixoto committed
331
332
333
    for v in g.vertices():
        if g.is_directed():
            for e in v.out_edges():
Tiago Peixoto's avatar
Tiago Peixoto committed
334
                m[index[v], eindex[e]] += -1
Tiago Peixoto's avatar
Tiago Peixoto committed
335
            for e in v.in_edges():
Tiago Peixoto's avatar
Tiago Peixoto committed
336
                m[index[v], eindex[e]] += 1
Tiago Peixoto's avatar
Tiago Peixoto committed
337
338
        else:
            for e in v.out_edges():
Tiago Peixoto's avatar
Tiago Peixoto committed
339
                m[index[v], eindex[e]] += 1
Tiago Peixoto's avatar
Tiago Peixoto committed
340
341
342
    if sparse:
        m = m.tocsr()
    return m