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"""!
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3
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@brief Neural Network: Oscillatory Neural Network based on Kuramoto model
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@details Based on article description:
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- A.Arenas, Y.Moreno, C.Zhou. Synchronization in complex networks. 2008.
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6
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- X.B.Lu. Adaptive Cluster Synchronization in Coupled Phase Oscillators. 2009.
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7
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- X.Lou. Adaptive Synchronizability of Coupled Oscillators With Switching. 2012.
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8
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- A.Novikov, E.Benderskaya. Oscillatory Neural Networks Based on the Kuramoto Model. 2014.
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10
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@authors Andrei Novikov ([email protected])
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@date 2014-2017
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@copyright GNU Public License
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14
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@cond GNU_PUBLIC_LICENSE
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PyClustering is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 3 of the License, or
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(at your option) any later version.
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PyClustering is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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@endcond
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"""
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import matplotlib.pyplot as plt;
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import matplotlib.animation as animation;
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import math;
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import numpy;
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import random;
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import pyclustering.core.sync_wrapper as wrapper;
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from scipy.integrate import odeint;
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41
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42
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from pyclustering.nnet import network, conn_represent, conn_type, initial_type, solve_type;
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from pyclustering.utils import pi, draw_dynamics, draw_dynamics_set, set_ax_param;
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45
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class sync_dynamic:
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"""!
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@brief Represents output dynamic of Sync.
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"""
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@property
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def output(self):
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54
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"""!
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@brief (list) Returns output dynamic of the Sync network (phase coordinates of each oscillator in the network) during simulation.
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56
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"""
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if ( (self._ccore_sync_dynamic_pointer is not None) and ( (self._dynamic is None) or (len(self._dynamic) == 0) ) ):
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self._dynamic = wrapper.sync_dynamic_get_output(self._ccore_sync_dynamic_pointer);
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return self._dynamic;
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@property
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def time(self):
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66
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"""!
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67
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@brief (list) Returns sampling times when dynamic is measured during simulation.
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68
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69
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"""
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if ( (self._ccore_sync_dynamic_pointer is not None) and ( (self._time is None) or (len(self._time) == 0) ) ):
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self._time = wrapper.sync_dynamic_get_time(self._ccore_sync_dynamic_pointer);
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return self._time;
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def __init__(self, phase, time, ccore = None):
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77
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"""!
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@brief Constructor of Sync dynamic.
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79
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80
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@param[in] phase (list): Dynamic of oscillators on each step of simulation. If ccore pointer is specified than it can be ignored.
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@param[in] time (list): Simulation time.
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@param[in] ccore (ctypes.pointer): Pointer to CCORE sync_dynamic instance in memory.
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"""
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self._dynamic = phase;
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self._time = time;
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self._ccore_sync_dynamic_pointer = ccore;
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def __del__(self):
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"""!
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@brief Default destructor of Sync dynamic.
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"""
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if (self._ccore_sync_dynamic_pointer is not None):
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wrapper.sync_dynamic_destroy(self._ccore_sync_dynamic_pointer);
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def __len__(self):
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101
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"""!
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@brief (uint) Returns number of simulation steps that are stored in dynamic.
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103
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"""
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if (self._ccore_sync_dynamic_pointer is not None):
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return wrapper.sync_dynamic_get_size(self._ccore_sync_dynamic_pointer);
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return len(self._dynamic);
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110
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def __getitem__(self, index):
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112
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"""!
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@brief Indexing of the dynamic.
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"""
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if (index is 0):
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return self.time;
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elif (index is 1):
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return self.output;
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else:
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raise NameError('Out of range ' + index + ': only indexes 0 and 1 are supported.');
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def allocate_sync_ensembles(self, tolerance = 0.01, indexes = None, iteration = None):
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127
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"""!
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128
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@brief Allocate clusters in line with ensembles of synchronous oscillators where each synchronous ensemble corresponds to only one cluster.
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130
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@param[in] tolerance (double): Maximum error for allocation of synchronous ensemble oscillators.
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@param[in] indexes (list): List of real object indexes and it should be equal to amount of oscillators (in case of 'None' - indexes are in range [0; amount_oscillators]).
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@param[in] iteration (uint): Iteration of simulation that should be used for allocation.
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134
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@return (list) Grours (lists) of indexes of synchronous oscillators.
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135
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For example [ [index_osc1, index_osc3], [index_osc2], [index_osc4, index_osc5] ].
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137
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"""
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138
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139
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if (self._ccore_sync_dynamic_pointer is not None):
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140
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ensembles = wrapper.sync_dynamic_allocate_sync_ensembles(self._ccore_sync_dynamic_pointer, tolerance, iteration);
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141
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142
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if (indexes is not None):
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143
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for ensemble in ensembles:
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for index in range(len(ensemble)):
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ensemble[index] = indexes[ ensemble[index] ];
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146
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147
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return ensembles;
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148
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149
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if ( (self._dynamic is None) or (len(self._dynamic) == 0) ):
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150
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return [];
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151
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152
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number_oscillators = len(self._dynamic[0]);
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153
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last_state = None;
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154
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155
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if (iteration is None):
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156
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last_state = self._dynamic[len(self._dynamic) - 1];
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else:
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last_state = self._dynamic[iteration];
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159
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160
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clusters = [];
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if (number_oscillators > 0):
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162
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clusters.append([0]);
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163
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164
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for i in range(1, number_oscillators, 1):
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cluster_allocated = False;
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166
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for cluster in clusters:
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167
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for neuron_index in cluster:
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168
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last_state_shifted = abs(last_state[i] - 2 * pi);
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169
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170
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if ( ( (last_state[i] < (last_state[neuron_index] + tolerance)) and (last_state[i] > (last_state[neuron_index] - tolerance)) ) or
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171
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( (last_state_shifted < (last_state[neuron_index] + tolerance)) and (last_state_shifted > (last_state[neuron_index] - tolerance)) ) ):
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172
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cluster_allocated = True;
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173
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174
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real_index = i;
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175
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if (indexes is not None):
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176
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real_index = indexes[i];
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177
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178
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cluster.append(real_index);
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179
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break;
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180
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181
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if (cluster_allocated == True):
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break;
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184
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if (cluster_allocated == False):
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185
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clusters.append([i]);
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186
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187
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return clusters;
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188
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190
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def allocate_phase_matrix(self, grid_width = None, grid_height = None, iteration = None):
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191
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"""!
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192
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@brief Returns 2D matrix of phase values of oscillators at the specified iteration of simulation.
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193
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@details User should ensure correct matrix sizes in line with following expression grid_width x grid_height that should be equal to
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194
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amount of oscillators otherwise exception is thrown. If grid_width or grid_height are not specified than phase matrix size
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195
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will by calculated automatically by square root.
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197
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@param[in] grid_width (uint): Width of the allocated matrix.
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198
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@param[in] grid_height (uint): Height of the allocated matrix.
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199
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@param[in] iteration (uint): Number of iteration of simulation for which correlation matrix should be allocated.
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200
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If iternation number is not specified, the last step of simulation is used for the matrix allocation.
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202
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@return (list) Phase value matrix of oscillators with size [number_oscillators x number_oscillators].
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203
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204
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"""
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205
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206
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output_dynamic = self.output;
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207
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208
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if ( (output_dynamic is None) or (len(output_dynamic) == 0) ):
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209
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return [];
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210
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211
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current_dynamic = output_dynamic[len(output_dynamic) - 1];
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212
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if (iteration is not None):
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213
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current_dynamic = output_dynamic[iteration];
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214
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215
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width_matrix = grid_width;
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216
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height_matrix = grid_height;
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217
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number_oscillators = len(current_dynamic);
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218
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if ( (width_matrix is None) or (height_matrix is None) ):
|
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219
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width_matrix = int(math.ceil(math.sqrt(number_oscillators)));
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220
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height_matrix = width_matrix;
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221
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222
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if (number_oscillators != width_matrix * height_matrix):
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223
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raise NameError("Impossible to allocate phase matrix with specified sizes, amout of neurons should be equal to grid_width * grid_height.");
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224
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225
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226
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phase_matrix = [ [ 0.0 for i in range(width_matrix) ] for j in range(height_matrix) ];
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227
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for i in range(height_matrix):
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228
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for j in range(width_matrix):
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229
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phase_matrix[i][j] = current_dynamic[j + i * width_matrix];
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230
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231
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return phase_matrix;
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232
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233
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234
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def allocate_correlation_matrix(self, iteration = None):
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235
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"""!
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236
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|
@brief Allocate correlation matrix between oscillators at the specified step of simulation.
|
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237
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|
238
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@param[in] iteration (uint): Number of iteration of simulation for which correlation matrix should be allocated.
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239
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If iternation number is not specified, the last step of simulation is used for the matrix allocation.
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240
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241
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@return (list) Correlation matrix between oscillators with size [number_oscillators x number_oscillators].
|
|
242
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243
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"""
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|
244
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|
245
|
|
|
if (self._ccore_sync_dynamic_pointer is not None):
|
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246
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return wrapper.sync_dynamic_allocate_correlation_matrix(self._ccore_sync_dynamic_pointer, iteration);
|
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247
|
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|
|
248
|
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|
if ( (self._dynamic is None) or (len(self._dynamic) == 0) ):
|
|
249
|
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|
return [];
|
|
250
|
|
|
|
|
251
|
|
|
dynamic = self._dynamic;
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252
|
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|
current_dynamic = dynamic[len(dynamic) - 1];
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253
|
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|
|
|
254
|
|
|
if (iteration is not None):
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|
255
|
|
|
current_dynamic = dynamic[iteration];
|
|
256
|
|
|
|
|
257
|
|
|
number_oscillators = len(dynamic[0]);
|
|
258
|
|
|
affinity_matrix = [ [ 0.0 for i in range(number_oscillators) ] for j in range(number_oscillators) ];
|
|
259
|
|
|
|
|
260
|
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|
for i in range(number_oscillators):
|
|
261
|
|
|
for j in range(number_oscillators):
|
|
262
|
|
|
phase1 = current_dynamic[i];
|
|
263
|
|
|
phase2 = current_dynamic[j];
|
|
264
|
|
|
|
|
265
|
|
|
affinity_matrix[i][j] = abs(math.sin(phase1 - phase2));
|
|
266
|
|
|
|
|
267
|
|
|
return affinity_matrix;
|
|
268
|
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|
269
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|
|
270
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|
def calculate_order_parameter(self, start_iteration = None, stop_iteration = None):
|
|
271
|
|
|
"""!
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|
|
|
|
|
|
272
|
|
|
@brief Calculates level of global synchorization (order parameter).
|
|
273
|
|
|
@details This parameter is tend 1.0 when the oscillatory network close to global synchronization and it tend to 0.0 when
|
|
274
|
|
|
desynchronization is observed in the network. Order parameter is calculated using following equation:
|
|
275
|
|
|
|
|
276
|
|
|
\f[
|
|
277
|
|
|
r_{c}=\frac{1}{Ne^{i\varphi }}\sum_{j=0}^{N}e^{i\theta_{j}};
|
|
278
|
|
|
\f]
|
|
279
|
|
|
|
|
280
|
|
|
where \f$\varphi\f$ is a average phase coordinate in the network, \f$N\f$ is an amount of oscillators in the network.
|
|
281
|
|
|
|
|
282
|
|
|
@param[in] start_iteration (uint): The first iteration that is used for calculation, if 'None' then the last iteration is used.
|
|
283
|
|
|
@param[in] stop_iteration (uint): The last iteration that is used for calculation, if 'None' then 'start_iteration' + 1 is used.
|
|
284
|
|
|
|
|
285
|
|
|
Example:
|
|
286
|
|
|
@code
|
|
287
|
|
|
oscillatory_network = sync(16, type_conn = conn_type.ALL_TO_ALL);
|
|
288
|
|
|
output_dynamic = oscillatory_network.simulate_static(100, 10);
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290
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print("Order parameter at the last step: ", output_dynamic.calculate_order_parameter());
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291
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print("Order parameter at the first step:", output_dynamic.calculate_order_parameter(0));
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292
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print("Order parameter evolution between 40 and 50 steps:", output_dynamic.calculate_order_parameter(40, 50));
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@endcode
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295
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@return (list) List of levels of global synchronization (order parameter evolution).
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297
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@see calculate_sync_order()
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299
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"""
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301
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if (start_iteration is None):
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start_iteration = len(self._dynamic) - 1;
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304
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if (stop_iteration is None):
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305
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stop_iteration = start_iteration + 1;
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306
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307
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sequence_order = [];
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308
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for index in range(start_iteration, stop_iteration):
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sequence_order.append(sync_dynamic.calculate_sync_order(self._dynamic[index]));
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311
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return sequence_order;
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314
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@staticmethod
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315
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def calculate_sync_order(oscillator_phases):
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316
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"""!
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317
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@brief Calculates level of global synchorization (order parameter) for input phases.
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318
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@details This parameter is tend 1.0 when the oscillatory network close to global synchronization and it tend to 0.0 when
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319
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desynchronization is observed in the network.
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320
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321
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@param[in] oscillator_phases (list): List of oscillator phases that are used for level of global synchronization.
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322
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323
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@return (double) Level of global synchronization (order parameter).
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324
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325
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@see calculate_order_parameter()
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326
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327
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"""
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328
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329
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exp_amount = 0.0;
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330
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average_phase = 0.0;
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331
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332
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for phase in oscillator_phases:
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333
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exp_amount += math.expm1( abs(1j * phase) );
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334
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average_phase += phase;
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335
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336
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exp_amount /= len(oscillator_phases);
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337
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|
average_phase = math.expm1( abs(1j * (average_phase / len(oscillator_phases))) );
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338
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339
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return abs(average_phase) / abs(exp_amount);
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340
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341
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342
|
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|
class sync_visualizer:
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343
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"""!
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344
|
|
|
@brief Visualizer of output dynamic of sync network (Sync).
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345
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346
|
|
|
"""
|
|
347
|
|
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|
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348
|
|
|
@staticmethod
|
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349
|
|
|
def show_output_dynamic(sync_output_dynamic):
|
|
350
|
|
|
"""!
|
|
351
|
|
|
@brief Shows output dynamic (output of each oscillator) during simulation.
|
|
352
|
|
|
|
|
353
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
354
|
|
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|
|
355
|
|
|
@see show_output_dynamics
|
|
356
|
|
|
|
|
357
|
|
|
"""
|
|
358
|
|
|
|
|
359
|
|
|
draw_dynamics(sync_output_dynamic.time, sync_output_dynamic.output, x_title = "t", y_title = "phase", y_lim = [0, 2 * 3.14]);
|
|
360
|
|
|
|
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361
|
|
|
|
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362
|
|
|
@staticmethod
|
|
363
|
|
|
def show_output_dynamics(sync_output_dynamics):
|
|
364
|
|
|
"""!
|
|
365
|
|
|
@brief Shows several output dynamics (output of each oscillator) during simulation.
|
|
366
|
|
|
@details Each dynamic is presented on separate plot.
|
|
367
|
|
|
|
|
368
|
|
|
@param[in] sync_output_dynamics (list): list of output dynamics 'sync_dynamic' of the Sync network.
|
|
369
|
|
|
|
|
370
|
|
|
@see show_output_dynamic
|
|
371
|
|
|
|
|
372
|
|
|
"""
|
|
373
|
|
|
|
|
374
|
|
|
draw_dynamics_set(sync_output_dynamics, "t", "phase", None, [0, 2 * 3.14], False, False);
|
|
375
|
|
|
|
|
376
|
|
|
|
|
377
|
|
|
@staticmethod
|
|
378
|
|
|
def show_correlation_matrix(sync_output_dynamic, iteration = None):
|
|
379
|
|
|
"""!
|
|
380
|
|
View Code Duplication |
@brief Shows correlation matrix between oscillators at the specified iteration.
|
|
|
|
|
|
|
381
|
|
|
|
|
382
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
383
|
|
|
@param[in] iteration (uint): Number of interation of simulation for which correlation matrix should be allocated.
|
|
384
|
|
|
If iternation number is not specified, the last step of simulation is used for the matrix allocation.
|
|
385
|
|
|
|
|
386
|
|
|
"""
|
|
387
|
|
|
|
|
388
|
|
|
_ = plt.figure();
|
|
389
|
|
|
correlation_matrix = sync_output_dynamic.allocate_correlation_matrix(iteration);
|
|
390
|
|
|
|
|
391
|
|
|
plt.imshow(correlation_matrix, cmap = plt.get_cmap('cool'), interpolation='kaiser', vmin = 0.0, vmax = 1.0);
|
|
392
|
|
|
plt.show();
|
|
393
|
|
|
|
|
394
|
|
|
|
|
395
|
|
|
@staticmethod
|
|
396
|
|
|
def show_phase_matrix(sync_output_dynamic, grid_width = None, grid_height = None, iteration = None):
|
|
397
|
|
|
"""!
|
|
398
|
|
|
@brief Shows 2D matrix of phase values of oscillators at the specified iteration.
|
|
399
|
|
|
@details User should ensure correct matrix sizes in line with following expression grid_width x grid_height that should be equal to
|
|
400
|
|
|
amount of oscillators otherwise exception is thrown. If grid_width or grid_height are not specified than phase matrix size
|
|
401
|
|
|
will by calculated automatically by square root.
|
|
402
|
|
|
|
|
403
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network whose phase matrix should be shown.
|
|
404
|
|
|
@param[in] grid_width (uint): Width of the phase matrix.
|
|
405
|
|
|
@param[in] grid_height (uint): Height of the phase matrix.
|
|
406
|
|
|
@param[in] iteration (uint): Number of iteration of simulation for which correlation matrix should be allocated.
|
|
407
|
|
|
If iternation number is not specified, the last step of simulation is used for the matrix allocation.
|
|
408
|
|
|
|
|
409
|
|
|
"""
|
|
410
|
|
|
|
|
411
|
|
|
_ = plt.figure();
|
|
412
|
|
|
phase_matrix = sync_output_dynamic.allocate_phase_matrix(grid_width, grid_height, iteration);
|
|
413
|
|
|
|
|
414
|
|
View Code Duplication |
plt.imshow(phase_matrix, cmap = plt.get_cmap('jet'), interpolation='kaiser', vmin = 0.0, vmax = 2.0 * math.pi);
|
|
|
|
|
|
|
415
|
|
|
plt.show();
|
|
416
|
|
|
|
|
417
|
|
|
|
|
418
|
|
|
@staticmethod
|
|
419
|
|
|
def show_order_parameter(sync_output_dynamic, start_iteration = None, stop_iteration = None):
|
|
420
|
|
|
"""!
|
|
421
|
|
|
@brief Shows evolution of order parameter (level of global synchronization in the network).
|
|
422
|
|
|
|
|
423
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network whose evolution of global synchronization should be visualized.
|
|
424
|
|
|
@param[in] start_iteration (uint): The first iteration that is used for calculation, if 'None' then the first is used
|
|
425
|
|
|
@param[in] stop_iteration (uint): The last iteration that is used for calculation, if 'None' then the last is used.
|
|
426
|
|
|
|
|
427
|
|
|
"""
|
|
428
|
|
|
|
|
429
|
|
|
if (start_iteration is None):
|
|
430
|
|
|
start_iteration = 0;
|
|
431
|
|
|
|
|
432
|
|
|
if (stop_iteration is None):
|
|
433
|
|
|
stop_iteration = len(sync_output_dynamic);
|
|
434
|
|
|
|
|
435
|
|
|
order_parameter = sync_output_dynamic.calculate_order_parameter(start_iteration, stop_iteration);
|
|
436
|
|
|
axis = plt.subplot(111);
|
|
437
|
|
|
plt.plot(sync_output_dynamic.time[start_iteration:stop_iteration], order_parameter, 'b-', linewidth = 2.0);
|
|
438
|
|
|
set_ax_param(axis, "t", "R (order parameter)", None, [0.0, 1.05]);
|
|
439
|
|
|
|
|
440
|
|
|
plt.show();
|
|
441
|
|
|
|
|
442
|
|
|
|
|
443
|
|
|
@staticmethod
|
|
444
|
|
|
def animate_output_dynamic(sync_output_dynamic, animation_velocity = 75, save_movie = None):
|
|
445
|
|
|
"""!
|
|
446
|
|
|
@brief Shows animation of output dynamic (output of each oscillator) during simulation on a circle from [0; 2pi].
|
|
447
|
|
|
|
|
448
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
449
|
|
|
@param[in] animation_velocity (uint): Interval between frames in milliseconds.
|
|
450
|
|
|
@param[in] save_movie (string): If it is specified then animation will be stored to file that is specified in this parameter.
|
|
451
|
|
|
|
|
452
|
|
|
"""
|
|
453
|
|
|
|
|
454
|
|
|
figure = plt.figure();
|
|
455
|
|
|
|
|
456
|
|
|
dynamic = sync_output_dynamic.output[0];
|
|
457
|
|
|
artist, = plt.polar(dynamic, [1.0] * len(dynamic), 'o', color = 'blue');
|
|
458
|
|
|
|
|
459
|
|
|
def init_frame():
|
|
460
|
|
|
return [ artist ];
|
|
461
|
|
|
|
|
462
|
|
|
def frame_generation(index_dynamic):
|
|
463
|
|
|
dynamic = sync_output_dynamic.output[index_dynamic];
|
|
464
|
|
|
artist.set_data(dynamic, [1.0] * len(dynamic));
|
|
465
|
|
|
|
|
466
|
|
|
return [ artist ];
|
|
467
|
|
|
|
|
468
|
|
|
phase_animation = animation.FuncAnimation(figure, frame_generation, len(sync_output_dynamic), interval = animation_velocity, init_func = init_frame, repeat_delay = 5000);
|
|
469
|
|
|
|
|
470
|
|
|
if (save_movie is not None):
|
|
471
|
|
|
phase_animation.save(save_movie, writer = 'ffmpeg', fps = 15, bitrate = 1500);
|
|
472
|
|
|
else:
|
|
473
|
|
|
plt.show();
|
|
474
|
|
|
|
|
475
|
|
|
|
|
476
|
|
|
@staticmethod
|
|
477
|
|
|
def animate_correlation_matrix(sync_output_dynamic, animation_velocity = 75, colormap = 'cool', save_movie = None):
|
|
478
|
|
|
"""!
|
|
479
|
|
|
@brief Shows animation of correlation matrix between oscillators during simulation.
|
|
480
|
|
|
|
|
481
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
482
|
|
|
@param[in] animation_velocity (uint): Interval between frames in milliseconds.
|
|
483
|
|
|
@param[in] colormap (string): Name of colormap that is used by matplotlib ('gray', 'pink', 'cool', spring', etc.).
|
|
484
|
|
|
@param[in] save_movie (string): If it is specified then animation will be stored to file that is specified in this parameter.
|
|
485
|
|
|
|
|
486
|
|
|
"""
|
|
487
|
|
|
|
|
488
|
|
|
figure = plt.figure();
|
|
489
|
|
|
|
|
490
|
|
|
correlation_matrix = sync_output_dynamic.allocate_correlation_matrix(0);
|
|
491
|
|
|
artist = plt.imshow(correlation_matrix, cmap = plt.get_cmap(colormap), interpolation='kaiser', hold = True, vmin = 0.0, vmax = 1.0);
|
|
492
|
|
|
|
|
493
|
|
|
def init_frame():
|
|
494
|
|
|
return [ artist ];
|
|
495
|
|
|
|
|
496
|
|
|
def frame_generation(index_dynamic):
|
|
497
|
|
|
correlation_matrix = sync_output_dynamic.allocate_correlation_matrix(index_dynamic);
|
|
498
|
|
|
artist.set_data(correlation_matrix);
|
|
499
|
|
|
|
|
500
|
|
|
return [ artist ];
|
|
501
|
|
|
|
|
502
|
|
|
correlation_animation = animation.FuncAnimation(figure, frame_generation, len(sync_output_dynamic), init_func = init_frame, interval = animation_velocity , repeat_delay = 1000, blit = True);
|
|
503
|
|
|
|
|
504
|
|
|
if (save_movie is not None):
|
|
505
|
|
|
correlation_animation.save(save_movie, writer = 'ffmpeg', fps = 15, bitrate = 1500);
|
|
506
|
|
|
else:
|
|
507
|
|
|
plt.show();
|
|
508
|
|
|
|
|
509
|
|
|
|
|
510
|
|
|
@staticmethod
|
|
511
|
|
|
def animate_phase_matrix(sync_output_dynamic, grid_width = None, grid_height = None, animation_velocity = 75, colormap = 'jet', save_movie = None):
|
|
512
|
|
|
"""!
|
|
513
|
|
|
@brief Shows animation of phase matrix between oscillators during simulation on 2D stage.
|
|
514
|
|
|
@details If grid_width or grid_height are not specified than phase matrix size will by calculated automatically by square root.
|
|
515
|
|
|
|
|
516
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
517
|
|
|
@param[in] grid_width (uint): Width of the phase matrix.
|
|
518
|
|
|
@param[in] grid_height (uint): Height of the phase matrix.
|
|
519
|
|
|
@param[in] animation_velocity (uint): Interval between frames in milliseconds.
|
|
520
|
|
|
@param[in] colormap (string): Name of colormap that is used by matplotlib ('gray', 'pink', 'cool', spring', etc.).
|
|
521
|
|
|
@param[in] save_movie (string): If it is specified then animation will be stored to file that is specified in this parameter.
|
|
522
|
|
|
|
|
523
|
|
|
"""
|
|
524
|
|
|
|
|
525
|
|
|
figure = plt.figure();
|
|
526
|
|
|
|
|
527
|
|
|
def init_frame():
|
|
528
|
|
|
return frame_generation(0);
|
|
529
|
|
|
|
|
530
|
|
|
def frame_generation(index_dynamic):
|
|
531
|
|
|
figure.clf();
|
|
532
|
|
|
axis = figure.add_subplot(111);
|
|
533
|
|
|
|
|
534
|
|
|
phase_matrix = sync_output_dynamic.allocate_phase_matrix(grid_width, grid_height, index_dynamic);
|
|
535
|
|
|
axis.imshow(phase_matrix, cmap = plt.get_cmap(colormap), interpolation='kaiser', vmin = 0.0, vmax = 2.0 * math.pi);
|
|
536
|
|
|
artist = figure.gca();
|
|
537
|
|
|
|
|
538
|
|
|
return [ artist ];
|
|
539
|
|
|
|
|
540
|
|
|
phase_animation = animation.FuncAnimation(figure, frame_generation, len(sync_output_dynamic), init_func = init_frame, interval = animation_velocity , repeat_delay = 1000);
|
|
541
|
|
|
|
|
542
|
|
|
if (save_movie is not None):
|
|
543
|
|
|
phase_animation.save(save_movie, writer = 'ffmpeg', fps = 15, bitrate = 1500);
|
|
544
|
|
|
else:
|
|
545
|
|
|
plt.show();
|
|
546
|
|
|
|
|
547
|
|
|
|
|
548
|
|
|
|
|
549
|
|
|
@staticmethod
|
|
550
|
|
|
def animate(sync_output_dynamic, title = None, save_movie = None):
|
|
551
|
|
|
"""!
|
|
552
|
|
|
@brief Shows animation of phase coordinates and animation of correlation matrix together for the Sync dynamic output on the same figure.
|
|
553
|
|
|
|
|
554
|
|
|
@param[in] sync_output_dynamic (sync_dynamic): Output dynamic of the Sync network.
|
|
555
|
|
|
@param[in] title (string): Title of the animation that is displayed on a figure if it is specified.
|
|
556
|
|
|
@param[in] save_movie (string): If it is specified then animation will be stored to file that is specified in this parameter.
|
|
557
|
|
|
|
|
558
|
|
|
"""
|
|
559
|
|
|
|
|
560
|
|
|
dynamic = sync_output_dynamic.output[0];
|
|
561
|
|
|
correlation_matrix = sync_output_dynamic.allocate_correlation_matrix(0);
|
|
562
|
|
|
|
|
563
|
|
|
figure = plt.figure(1);
|
|
564
|
|
|
if (title is not None):
|
|
565
|
|
|
figure.suptitle(title, fontsize = 26, fontweight = 'bold')
|
|
566
|
|
|
|
|
567
|
|
|
ax1 = figure.add_subplot(121, projection='polar');
|
|
568
|
|
|
ax2 = figure.add_subplot(122);
|
|
569
|
|
|
|
|
570
|
|
|
artist1, = ax1.plot(dynamic, [1.0] * len(dynamic), marker = 'o', color = 'blue', ls = '');
|
|
571
|
|
|
artist2 = ax2.imshow(correlation_matrix, cmap = plt.get_cmap('Accent'), interpolation='kaiser');
|
|
572
|
|
|
|
|
573
|
|
|
def init_frame():
|
|
574
|
|
|
return [ artist1, artist2 ];
|
|
575
|
|
|
|
|
576
|
|
|
def frame_generation(index_dynamic):
|
|
577
|
|
|
dynamic = sync_output_dynamic.output[index_dynamic];
|
|
578
|
|
|
artist1.set_data(dynamic, [1.0] * len(dynamic));
|
|
579
|
|
|
|
|
580
|
|
|
correlation_matrix = sync_output_dynamic.allocate_correlation_matrix(index_dynamic);
|
|
581
|
|
|
artist2.set_data(correlation_matrix);
|
|
582
|
|
|
|
|
583
|
|
|
return [ artist1, artist2 ];
|
|
584
|
|
|
|
|
585
|
|
|
dynamic_animation = animation.FuncAnimation(figure, frame_generation, len(sync_output_dynamic), interval = 75, init_func = init_frame, repeat_delay = 5000);
|
|
586
|
|
|
|
|
587
|
|
|
if (save_movie is not None):
|
|
588
|
|
|
dynamic_animation.save(save_movie, writer = 'ffmpeg', fps = 15, bitrate = 1500);
|
|
589
|
|
|
else:
|
|
590
|
|
|
plt.show();
|
|
591
|
|
|
|
|
592
|
|
|
|
|
593
|
|
|
|
|
594
|
|
|
class sync_network(network):
|
|
595
|
|
|
"""!
|
|
596
|
|
|
@brief Model of oscillatory network that is based on the Kuramoto model of synchronization.
|
|
597
|
|
|
|
|
598
|
|
|
"""
|
|
599
|
|
|
|
|
600
|
|
|
def __init__(self, num_osc, weight = 1, frequency = 0, type_conn = conn_type.ALL_TO_ALL, representation = conn_represent.MATRIX, initial_phases = initial_type.RANDOM_GAUSSIAN, ccore = False):
|
|
601
|
|
|
"""!
|
|
602
|
|
|
@brief Constructor of oscillatory network is based on Kuramoto model.
|
|
603
|
|
|
|
|
604
|
|
|
@param[in] num_osc (uint): Number of oscillators in the network.
|
|
605
|
|
|
@param[in] weight (double): Coupling strength of the links between oscillators.
|
|
606
|
|
|
@param[in] frequency (double): Multiplier of internal frequency of the oscillators.
|
|
607
|
|
|
@param[in] type_conn (conn_type): Type of connection between oscillators in the network (all-to-all, grid, bidirectional list, etc.).
|
|
608
|
|
|
@param[in] representation (conn_represent): Internal representation of connection in the network: matrix or list.
|
|
609
|
|
|
@param[in] initial_phases (initial_type): Type of initialization of initial phases of oscillators (random, uniformly distributed, etc.).
|
|
610
|
|
|
@param[in] ccore (bool): If True simulation is performed by CCORE library (C++ implementation of pyclustering).
|
|
611
|
|
|
|
|
612
|
|
|
"""
|
|
613
|
|
|
|
|
614
|
|
|
self._ccore_network_pointer = None; # Pointer to CCORE Sync implementation of the network.
|
|
615
|
|
|
|
|
616
|
|
|
if (ccore is True):
|
|
617
|
|
|
self._ccore_network_pointer = wrapper.sync_create_network(num_osc, weight, frequency, type_conn, initial_phases);
|
|
618
|
|
|
else:
|
|
619
|
|
|
super().__init__(num_osc, type_conn, representation);
|
|
620
|
|
|
|
|
621
|
|
|
self._weight = weight;
|
|
622
|
|
|
|
|
623
|
|
|
self._phases = list();
|
|
624
|
|
|
self._freq = list();
|
|
625
|
|
|
|
|
626
|
|
|
random.seed();
|
|
627
|
|
|
for index in range(0, num_osc, 1):
|
|
628
|
|
|
if (initial_phases == initial_type.RANDOM_GAUSSIAN):
|
|
629
|
|
|
self._phases.append(random.random() * 2 * pi);
|
|
630
|
|
|
|
|
631
|
|
|
elif (initial_phases == initial_type.EQUIPARTITION):
|
|
632
|
|
|
self._phases.append( pi / num_osc * index);
|
|
633
|
|
|
|
|
634
|
|
|
self._freq.append(random.random() * frequency);
|
|
635
|
|
|
|
|
636
|
|
|
|
|
637
|
|
|
def __del__(self):
|
|
638
|
|
|
"""!
|
|
639
|
|
|
@brief Destructor of oscillatory network is based on Kuramoto model.
|
|
640
|
|
|
|
|
641
|
|
|
"""
|
|
642
|
|
|
|
|
643
|
|
|
if (self._ccore_network_pointer is not None):
|
|
644
|
|
|
wrapper.sync_destroy_network(self._ccore_network_pointer);
|
|
645
|
|
|
self._ccore_network_pointer = None;
|
|
646
|
|
|
|
|
647
|
|
|
|
|
648
|
|
|
def sync_order(self):
|
|
649
|
|
|
"""!
|
|
|
|
|
|
|
650
|
|
|
@brief Calculates current level of global synchorization (order parameter) in the network.
|
|
651
|
|
|
@details This parameter is tend 1.0 when the oscillatory network close to global synchronization and it tend to 0.0 when
|
|
652
|
|
|
desynchronization is observed in the network. Order parameter is calculated using following equation:
|
|
653
|
|
|
|
|
654
|
|
|
\f[
|
|
655
|
|
|
r_{c}=\frac{1}{Ne^{i\varphi }}\sum_{j=0}^{N}e^{i\theta_{j}};
|
|
656
|
|
|
\f]
|
|
657
|
|
|
|
|
658
|
|
|
where \f$\varphi\f$ is a average phase coordinate in the network, \f$N\f$ is an amount of oscillators in the network.
|
|
659
|
|
|
|
|
660
|
|
|
Example:
|
|
661
|
|
|
@code
|
|
662
|
|
|
oscillatory_network = sync(16, type_conn = conn_type.ALL_TO_ALL);
|
|
663
|
|
|
output_dynamic = oscillatory_network.simulate_static(100, 10);
|
|
664
|
|
|
|
|
665
|
|
|
if (oscillatory_network.sync_order() < 0.9): print("Global synchronization is not reached yet.");
|
|
666
|
|
|
else: print("Global synchronization is reached.");
|
|
667
|
|
|
@endcode
|
|
668
|
|
|
|
|
669
|
|
|
@return (double) Level of global synchronization (order parameter).
|
|
670
|
|
|
|
|
671
|
|
|
@see sync_local_order()
|
|
672
|
|
|
|
|
673
|
|
|
"""
|
|
674
|
|
|
|
|
675
|
|
|
if (self._ccore_network_pointer is not None):
|
|
676
|
|
|
return wrapper.sync_order(self._ccore_network_pointer);
|
|
677
|
|
|
|
|
678
|
|
|
return sync_dynamic.calculate_sync_order(self._phases);
|
|
679
|
|
|
|
|
680
|
|
|
|
|
681
|
|
|
def sync_local_order(self):
|
|
682
|
|
|
"""!
|
|
683
|
|
|
@brief Calculates current level of local (partial) synchronization in the network.
|
|
684
|
|
|
|
|
685
|
|
|
@return (double) Level of local (partial) synchronization.
|
|
686
|
|
|
|
|
687
|
|
|
@see sync_order()
|
|
688
|
|
|
|
|
689
|
|
|
"""
|
|
690
|
|
|
|
|
691
|
|
|
if (self._ccore_network_pointer is not None):
|
|
692
|
|
|
return wrapper.sync_local_order(self._ccore_network_pointer);
|
|
693
|
|
|
|
|
694
|
|
|
exp_amount = 0.0;
|
|
695
|
|
|
num_neigh = 0;
|
|
696
|
|
|
|
|
697
|
|
|
for i in range(0, self._num_osc, 1):
|
|
698
|
|
|
for j in range(0, self._num_osc, 1):
|
|
699
|
|
|
if (self.has_connection(i, j) == True):
|
|
700
|
|
|
exp_amount += math.exp(-abs(self._phases[j] - self._phases[i]));
|
|
701
|
|
|
num_neigh += 1;
|
|
702
|
|
|
|
|
703
|
|
|
if (num_neigh == 0):
|
|
704
|
|
|
num_neigh = 1;
|
|
705
|
|
|
|
|
706
|
|
|
return exp_amount / num_neigh;
|
|
707
|
|
|
|
|
708
|
|
|
|
|
709
|
|
|
def _phase_kuramoto(self, teta, t, argv):
|
|
|
|
|
|
|
710
|
|
|
"""!
|
|
711
|
|
|
@brief Returns result of phase calculation for specified oscillator in the network.
|
|
712
|
|
|
|
|
713
|
|
|
@param[in] teta (double): Phase of the oscillator that is differentiated.
|
|
714
|
|
|
@param[in] t (double): Current time of simulation.
|
|
715
|
|
|
@param[in] argv (tuple): Index of the oscillator in the list.
|
|
716
|
|
|
|
|
717
|
|
|
@return (double) New phase for specified oscillator (don't assign here).
|
|
718
|
|
|
|
|
719
|
|
|
"""
|
|
720
|
|
|
|
|
721
|
|
|
index = argv;
|
|
722
|
|
|
phase = 0;
|
|
723
|
|
|
for k in range(0, self._num_osc):
|
|
724
|
|
|
if (self.has_connection(index, k) == True):
|
|
725
|
|
|
phase += math.sin(self._phases[k] - teta);
|
|
726
|
|
|
|
|
727
|
|
|
return ( self._freq[index] + (phase * self._weight / self._num_osc) );
|
|
728
|
|
|
|
|
729
|
|
|
|
|
730
|
|
|
def simulate(self, steps, time, solution = solve_type.FAST, collect_dynamic = True):
|
|
731
|
|
|
"""!
|
|
732
|
|
|
@brief Performs static simulation of Sync oscillatory network.
|
|
733
|
|
|
|
|
734
|
|
|
@param[in] steps (uint): Number steps of simulations during simulation.
|
|
735
|
|
|
@param[in] time (double): Time of simulation.
|
|
736
|
|
|
@param[in] solution (solve_type): Type of solution (solving).
|
|
737
|
|
|
@param[in] collect_dynamic (bool): If True - returns whole dynamic of oscillatory network, otherwise returns only last values of dynamics.
|
|
738
|
|
|
|
|
739
|
|
|
@return (list) Dynamic of oscillatory network. If argument 'collect_dynamic' = True, than return dynamic for the whole simulation time,
|
|
740
|
|
|
otherwise returns only last values (last step of simulation) of dynamic.
|
|
741
|
|
|
|
|
742
|
|
|
@see simulate_dynamic()
|
|
743
|
|
|
@see simulate_static()
|
|
744
|
|
|
|
|
745
|
|
|
"""
|
|
746
|
|
|
|
|
747
|
|
|
return self.simulate_static(steps, time, solution, collect_dynamic);
|
|
748
|
|
|
|
|
749
|
|
|
|
|
750
|
|
|
def simulate_dynamic(self, order = 0.998, solution = solve_type.FAST, collect_dynamic = False, step = 0.1, int_step = 0.01, threshold_changes = 0.0000001):
|
|
751
|
|
|
"""!
|
|
752
|
|
|
@brief Performs dynamic simulation of the network until stop condition is not reached. Stop condition is defined by input argument 'order'.
|
|
753
|
|
|
|
|
754
|
|
|
@param[in] order (double): Order of process synchronization, distributed 0..1.
|
|
755
|
|
|
@param[in] solution (solve_type): Type of solution.
|
|
756
|
|
|
@param[in] collect_dynamic (bool): If True - returns whole dynamic of oscillatory network, otherwise returns only last values of dynamics.
|
|
757
|
|
|
@param[in] step (double): Time step of one iteration of simulation.
|
|
758
|
|
|
@param[in] int_step (double): Integration step, should be less than step.
|
|
759
|
|
|
@param[in] threshold_changes (double): Additional stop condition that helps prevent infinite simulation, defines limit of changes of oscillators between current and previous steps.
|
|
760
|
|
|
|
|
761
|
|
|
@return (list) Dynamic of oscillatory network. If argument 'collect_dynamic' = True, than return dynamic for the whole simulation time,
|
|
762
|
|
|
otherwise returns only last values (last step of simulation) of dynamic.
|
|
763
|
|
|
|
|
764
|
|
|
@see simulate()
|
|
765
|
|
|
@see simulate_static()
|
|
766
|
|
|
|
|
767
|
|
|
"""
|
|
768
|
|
|
|
|
769
|
|
|
if (self._ccore_network_pointer is not None):
|
|
770
|
|
|
ccore_instance_dynamic = wrapper.sync_simulate_dynamic(self._ccore_network_pointer, order, solution, collect_dynamic, step, int_step, threshold_changes);
|
|
771
|
|
|
return sync_dynamic(None, None, ccore_instance_dynamic);
|
|
772
|
|
|
|
|
773
|
|
|
# For statistics and integration
|
|
774
|
|
|
time_counter = 0;
|
|
775
|
|
|
|
|
776
|
|
|
# Prevent infinite loop. It's possible when required state cannot be reached.
|
|
777
|
|
|
previous_order = 0;
|
|
778
|
|
|
current_order = self.sync_local_order();
|
|
779
|
|
|
|
|
780
|
|
|
# If requested input dynamics
|
|
781
|
|
|
dyn_phase = [];
|
|
782
|
|
|
dyn_time = [];
|
|
783
|
|
|
if (collect_dynamic == True):
|
|
784
|
|
|
dyn_phase.append(self._phases);
|
|
785
|
|
|
dyn_time.append(0);
|
|
786
|
|
|
|
|
787
|
|
|
# Execute until sync state will be reached
|
|
788
|
|
|
while (current_order < order):
|
|
789
|
|
|
# update states of oscillators
|
|
790
|
|
|
self._phases = self._calculate_phases(solution, time_counter, step, int_step);
|
|
791
|
|
|
|
|
792
|
|
|
# update time
|
|
793
|
|
|
time_counter += step;
|
|
794
|
|
|
|
|
795
|
|
|
# if requested input dynamic
|
|
796
|
|
|
if (collect_dynamic == True):
|
|
797
|
|
|
dyn_phase.append(self._phases);
|
|
798
|
|
|
dyn_time.append(time_counter);
|
|
799
|
|
|
|
|
800
|
|
|
# update orders
|
|
801
|
|
|
previous_order = current_order;
|
|
802
|
|
|
current_order = self.sync_local_order();
|
|
803
|
|
|
|
|
804
|
|
|
# hang prevention
|
|
805
|
|
|
if (abs(current_order - previous_order) < threshold_changes):
|
|
806
|
|
|
# print("Warning: sync_network::simulate_dynamic - simulation is aborted due to low level of convergence rate (order = " + str(current_order) + ").");
|
|
807
|
|
|
break;
|
|
808
|
|
|
|
|
809
|
|
|
if (collect_dynamic != True):
|
|
810
|
|
|
dyn_phase.append(self._phases);
|
|
811
|
|
|
dyn_time.append(time_counter);
|
|
812
|
|
|
|
|
813
|
|
|
output_sync_dynamic = sync_dynamic(dyn_phase, dyn_time, None);
|
|
814
|
|
|
return output_sync_dynamic;
|
|
815
|
|
|
|
|
816
|
|
|
|
|
817
|
|
|
def simulate_static(self, steps, time, solution = solve_type.FAST, collect_dynamic = False):
|
|
818
|
|
|
"""!
|
|
819
|
|
|
@brief Performs static simulation of oscillatory network.
|
|
820
|
|
|
|
|
821
|
|
|
@param[in] steps (uint): Number steps of simulations during simulation.
|
|
822
|
|
|
@param[in] time (double): Time of simulation.
|
|
823
|
|
|
@param[in] solution (solve_type): Type of solution.
|
|
824
|
|
|
@param[in] collect_dynamic (bool): If True - returns whole dynamic of oscillatory network, otherwise returns only last values of dynamics.
|
|
825
|
|
|
|
|
826
|
|
|
@return (list) Dynamic of oscillatory network. If argument 'collect_dynamic' = True, than return dynamic for the whole simulation time,
|
|
827
|
|
|
otherwise returns only last values (last step of simulation) of dynamic.
|
|
828
|
|
|
|
|
829
|
|
|
@see simulate()
|
|
830
|
|
|
@see simulate_dynamic()
|
|
831
|
|
|
|
|
832
|
|
|
"""
|
|
833
|
|
|
|
|
834
|
|
|
if (self._ccore_network_pointer is not None):
|
|
835
|
|
|
ccore_instance_dynamic = wrapper.sync_simulate_static(self._ccore_network_pointer, steps, time, solution, collect_dynamic);
|
|
836
|
|
|
return sync_dynamic(None, None, ccore_instance_dynamic);
|
|
837
|
|
|
|
|
838
|
|
|
dyn_phase = [];
|
|
839
|
|
|
dyn_time = [];
|
|
840
|
|
|
|
|
841
|
|
|
if (collect_dynamic == True):
|
|
842
|
|
|
dyn_phase.append(self._phases);
|
|
843
|
|
|
dyn_time.append(0);
|
|
844
|
|
|
|
|
845
|
|
|
step = time / steps;
|
|
846
|
|
|
int_step = step / 10.0;
|
|
847
|
|
|
|
|
848
|
|
|
for t in numpy.arange(step, time + step, step):
|
|
849
|
|
|
# update states of oscillators
|
|
850
|
|
|
self._phases = self._calculate_phases(solution, t, step, int_step);
|
|
851
|
|
|
|
|
852
|
|
|
# update states of oscillators
|
|
853
|
|
|
if (collect_dynamic == True):
|
|
854
|
|
|
dyn_phase.append(self._phases);
|
|
855
|
|
|
dyn_time.append(t);
|
|
856
|
|
|
|
|
857
|
|
|
if (collect_dynamic != True):
|
|
858
|
|
|
dyn_phase.append(self._phases);
|
|
859
|
|
|
dyn_time.append(time);
|
|
860
|
|
|
|
|
861
|
|
|
output_sync_dynamic = sync_dynamic(dyn_phase, dyn_time);
|
|
862
|
|
|
return output_sync_dynamic;
|
|
863
|
|
|
|
|
864
|
|
|
|
|
865
|
|
|
def _calculate_phases(self, solution, t, step, int_step):
|
|
866
|
|
|
"""!
|
|
867
|
|
|
@brief Calculates new phases for oscillators in the network in line with current step.
|
|
868
|
|
|
|
|
869
|
|
|
@param[in] solution (solve_type): Type solver of the differential equation.
|
|
870
|
|
|
@param[in] t (double): Time of simulation.
|
|
871
|
|
|
@param[in] step (double): Step of solution at the end of which states of oscillators should be calculated.
|
|
872
|
|
|
@param[in] int_step (double): Step differentiation that is used for solving differential equation.
|
|
873
|
|
|
|
|
874
|
|
|
@return (list) New states (phases) for oscillators.
|
|
875
|
|
|
|
|
876
|
|
|
"""
|
|
877
|
|
|
|
|
878
|
|
|
next_phases = [0.0] * self._num_osc; # new oscillator _phases
|
|
879
|
|
|
|
|
880
|
|
|
for index in range (0, self._num_osc, 1):
|
|
881
|
|
|
if (solution == solve_type.FAST):
|
|
882
|
|
|
result = self._phases[index] + self._phase_kuramoto(self._phases[index], 0, index);
|
|
883
|
|
|
next_phases[index] = self._phase_normalization(result);
|
|
884
|
|
|
|
|
885
|
|
|
elif (solution == solve_type.RK4):
|
|
886
|
|
|
result = odeint(self._phase_kuramoto, self._phases[index], numpy.arange(t - step, t, int_step), (index , ));
|
|
887
|
|
|
next_phases[index] = self._phase_normalization(result[len(result) - 1][0]);
|
|
888
|
|
|
|
|
889
|
|
|
else:
|
|
890
|
|
|
raise NameError("Solver '" + solution + "' is not supported");
|
|
891
|
|
|
|
|
892
|
|
|
return next_phases;
|
|
893
|
|
|
|
|
894
|
|
|
|
|
895
|
|
|
def _phase_normalization(self, teta):
|
|
896
|
|
|
"""!
|
|
897
|
|
|
@brief Normalization of phase of oscillator that should be placed between [0; 2 * pi].
|
|
898
|
|
|
|
|
899
|
|
|
@param[in] teta (double): phase of oscillator.
|
|
900
|
|
|
|
|
901
|
|
|
@return (double) Normalized phase.
|
|
902
|
|
|
|
|
903
|
|
|
"""
|
|
904
|
|
|
|
|
905
|
|
|
norm_teta = teta;
|
|
906
|
|
|
while (norm_teta > (2.0 * pi)) or (norm_teta < 0):
|
|
907
|
|
|
if (norm_teta > (2.0 * pi)):
|
|
908
|
|
|
norm_teta -= 2.0 * pi;
|
|
909
|
|
|
else:
|
|
910
|
|
|
norm_teta += 2.0 * pi;
|
|
911
|
|
|
|
|
912
|
|
|
return norm_teta;
|
|
913
|
|
|
|
annoviko /
pyclustering
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