__init__.py 8.49 KB
 Tiago Peixoto committed Sep 05, 2009 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 #! /usr/bin/env python # graph_tool.py -- a general graph manipulation python module # # Copyright (C) 2007 Tiago de Paula Peixoto # # 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 . """ graph_tool.spectral - Spectral properties ---------------------------------------------  Tiago Peixoto committed Oct 05, 2009 22 23 24 25 26 27 28 29 30 31 32 33 34  Summary +++++++ .. autosummary:: :nosignatures: adjacency laplacian incidence Contents ++++++++  Tiago Peixoto committed Sep 05, 2009 35 36 37 38 39 40 41 42 43 44 """ 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):  Tiago Peixoto committed Sep 07, 2009 45 46 47 48 49 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 81 82  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()  Tiago Peixoto committed Feb 20, 2010 83 84  [[ 0. 0. 0. ..., 0. 0. 0.] [ 0. 0. 0. ..., 0. 0. 0.]  Tiago Peixoto committed Sep 07, 2009 85 86  [ 0. 0. 0. ..., 0. 0. 0.] ...,  Tiago Peixoto committed Feb 20, 2010 87  [ 0. 0. 0. ..., 0. 1. 1.]  Tiago Peixoto committed Sep 07, 2009 88 89 90 91 92  [ 0. 0. 0. ..., 0. 0. 0.] [ 0. 0. 0. ..., 0. 0. 0.]] References ----------  Tiago Peixoto committed Oct 05, 2009 93  .. [wikipedia-adjacency] http://en.wikipedia.org/wiki/Adjacency_matrix  Tiago Peixoto committed Sep 07, 2009 94 95  """  Tiago Peixoto committed Sep 05, 2009 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133  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):  Tiago Peixoto committed Sep 07, 2009 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188  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()  Tiago Peixoto committed Feb 20, 2010 189  [[ 1. 0. 0. ..., 0. 0. 0. ]  Tiago Peixoto committed Sep 07, 2009 190  [ 0. 1. 0. ..., 0. 0. 0. ]  Tiago Peixoto committed Feb 20, 2010 191  [ 0. 0. 1. ..., 0. 0. 0. ]  Tiago Peixoto committed Sep 07, 2009 192  ...,  Tiago Peixoto committed Feb 20, 2010 193  [ 0. 0. 0. ..., 1. 0.05 0.05]  Tiago Peixoto committed Sep 07, 2009 194 195 196 197 198  [ 0. 0. 0. ..., 0. 1. 0. ] [ 0. 0. 0. ..., 0. 0. 1. ]] References ----------  Tiago Peixoto committed Oct 05, 2009 199  .. [wikipedia-laplacian] http://en.wikipedia.org/wiki/Laplacian_matrix  Tiago Peixoto committed Sep 07, 2009 200 201  """  Tiago Peixoto committed Sep 05, 2009 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230  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):  Tiago Peixoto committed Sep 07, 2009 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276  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.]  Tiago Peixoto committed Feb 20, 2010 277  [ 0. 0. 0. ..., 0. 0. 1.]  Tiago Peixoto committed Sep 07, 2009 278 279 280 281 282 283 284  ..., [ 0. 0. 0. ..., 0. 0. 0.] [ 0. 0. 0. ..., 0. 0. 0.] [ 0. 0. 0. ..., 0. 0. 0.]] References ----------  Tiago Peixoto committed Oct 05, 2009 285  .. [wikipedia-incidence] http://en.wikipedia.org/wiki/Incidence_matrix  Tiago Peixoto committed Sep 07, 2009 286 287  """  Tiago Peixoto committed Sep 05, 2009 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316  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