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第326行： − 这里分别针对上述状态空间和变量空间的因果涌现实例，提供了计算微观、宏观有效信息的python代码，以供参考。+ − 基于状态空间的计算需要：给定微观状态的概率转移矩阵，以及将微观状态对应到宏观状态的粗粒化方案；基于变量空间的计算需要指定微观布尔机制（是否带有噪声），微观元素个数，每个微观元素的输入元素，粗粒化方案包括元素分组方式和二元状态的分组方式。详见代码中示例。+ − − python:<syntaxhighlight lang="python3"> − import torch − import itertools − − # 计算信息熵 − def Entropia(a): − b = torch.nonzero(a > 0).squeeze(1) − return -torch.sum(a[b]*torch.log2(a[b])) − − # 基于状态概率矩阵计算有效信息 − def effective_information(G1): − """ − Calculate EI value of transition probability matrix − G1: TPM matrix − return the EI value − """ − N = G1.shape[0] − G = torch.zeros_like(G1) − nozero_number = 0 − for i in range(N): − if torch.sum(G1[i]) != 0: − G[i] = G1[i] / torch.sum(G1[i]) − nozero_number += 1 − Wout = torch.zeros(N) − Win = torch.zeros(N) − − for i in range(N): − if torch.sum(G[i]) != 0: − Wout[i] = torch.tensor(Entropia(G[i])) − − for i in range(N): − for j in range(N): − if G[i][j] != 0: − Win[j] += G[i][j] / nozero_number − Wout_average = torch.sum(Wout) / nozero_number − Win_entropy = Entropia(Win) − return Win_entropy - Wout_average − − − # 状态空间的宏观TPM生成，以及有效信息计算 − def calc_tpm(micro_tpm, group): − """ − Parameters − ---------- − micro_tpm: the transition probability matrix of system in micro level. − group: the coarse-graining methods (e.g., group = [(0, 1, 2), (3,)]: the micro states 0,1,2 are grouped into macro state 0, and the micro state 3 is grouped into macro state 1. − − Returns − ------- − micro_ei, macro_ei: the EI value of micro, macro TPM. − macro_tpm: the transition probability matrix of system in macro level − − − """ − # the number of micro/macro states − Nmi = micro_tpm.shape[0] − Nma = len(group) − − # generate the mapping matrix from micro to macro − mapping = torch.zeros((Nmi, Nma), dtype=torch.float) − for j in range(Nma): − for i in group[j]: − mapping[i][j] = 1 − − # apply the mapping matrix to micro_tpm, and get the macro_tpm − macro_tpm = mapping.t() @ micro_tpm @ mapping − − # calculate the EI of micro_tpm, macro_tpm − micro_ei = effective_information(micro_tpm) − macro_ei = effective_information(macro_tpm) − − return micro_ei, macro_ei, macro_tpm − − − # 变量空间的微观、宏观TPM生成，以及有效信息计算 − def calc_bn_ei(micro_mech, micro, edges, elem_group, mech_group): − """ − Calculate the effective information of boolean network in micro and macro level, given the micro mechanism and − coarse-graining methods. − − Parameters − ---------- − micro_mech: boolean function, e.g., {'00': [0.7, 0.3], '01': [0.7, 0.3], '10': [0.7, 0.3], '11': [0, 1]} the − value means the probability of transferring to '0' and '1', correspondingly. − micro: the number of micro elements − edges: a list of tuple. the list means the input micro elements. e.g., edges[1] = (2, 3) means − the input of elements[1] is [2] and [3] − elem_group: the coarse-graining methods of micro elements, e.g., [(0, 1), (2, 3)] means coarse-graining − elements[0],[1] to be a macro element, and the [2] and [3] to be another. − mech_group: the coarse-graining method of micro_mech, e.g., mech_group = {'00': '0', '01': '0', '10': '0', − '11': '1'} − − Returns − ------- − micro_ei, macro_ei: the EI value of micro, macro level − S_m, S_M: transition probability matrix of micro, macro level − − """ − # generate the statue list of boolean nodes, based on the number of micro elements − Nmi = 2 ** micro − node_list = list(itertools.product(['0', '1'], repeat=micro)) − S_m = torch.zeros((Nmi, Nmi), dtype=torch.float) − − # calculate the transition probability matrix of micro level − for i, pre in enumerate(node_list): − for j, tmp in enumerate(node_list): − tr_pr = 1 − for z in range(micro): − srce1 = pre[edges[z][0]] − srce2 = pre[edges[z][1]] − tr_pr *= micro_mech[srce1 + srce2][int(tmp[z])] − S_m[i][j] = tr_pr − − # calculate the grouping matrix from micro to macro level, based on the elem_group and mech_group. − unique = set(mech_group.values()) − Nma = len(unique) ** len(elem_group) − group = torch.zeros((Nmi, Nma), dtype=torch.float) − for i, item in enumerate(node_list): − Sma = '' − for elem in elem_group: − Smi = ''.join(item[idx] for idx in elem) − Sma += mech_group[Smi] − group[i, int(Sma, base=len(unique))] = 1 − − # calculate the transition probability matrix of macro level − S_M = group.t() @ S_m @ group − S_M = S_M / torch.sum(S_M, axis=1).unsqueeze(1) − − # calculate the EI of micro and macro level − micro_ei = effective_information(S_m) − macro_ei = effective_information(S_M) − − return micro_ei, macro_ei, S_m, S_M − − # Example：状态空间计算实例 − micro_tpm = torch.tensor([[1/3, 1/3, 1/3, 0], [1/3, 1/3, 1/3, 0], [1/3, 1/3, 1/3, 0], [0, 0, 0, 1]]) − group = [(0, 1, 2), (3,)] − micro_ei, macro_ei, macro_tpm = calc_tpm(micro_tpm, group) − CE = macro_ei - micro_ei − − # Example：提高确定性实例 − micro_mech = {'00': [0.7, 0.3], '01': [0.7, 0.3], '10': [0.7, 0.3], '11': [0, 1]} − micro = 4 − edges = [(2, 3), (2, 3), (0, 1), (0, 1)] − elem_group = [(0, 1), (2, 3)] − mech_group = {'00': '0', '01': '0', '10': '0', '11': '1'} − micro_ei, macro_ei, S_m, S_M = calc_bn_ei(micro_mech, micro, edges, elem_group, mech_group) − CE = macro_ei - micro_ei − </syntaxhighlight> − − 以上是给定粗粒化方案的有效信息和因果涌现计算。一些复杂网络的粗粒化方案通常不是很容易看出，可以基于算法来得到有效信息更大的粗粒化方案，例如贪婪算法，谱分解算法，算法实现步骤拆解可参考[[复杂网络中的因果涌现]]（代码来源：https://github.com/jkbren/einet）。 − − python:<syntaxhighlight lang="python3"> − # 需要在Github下载ce_net.py: https://github.com/jkbren/einet − from ce_net import * − − # directly define the micro TPM, or use the generating methods above in function calc_bn_ei. − micro_tpm = np.array([[1/3, 1/3, 1/3, 0], [1/3, 1/3, 1/3, 0], [1/3, 1/3, 1/3, 0], [0, 0, 0, 1]]) − − # check_network: transfer the TPM numpy ndarray format to digraph − G = check_network(micro_tpm) − − # causal_emergence: the function defined in ce_net.py − # optimizing the coarse-graining methods using greedy algorithm. − CE = causal_emergence(G) − # optimizing the coarse-graining methods using spectral algorithm. − CE = causal_emergence_spectral(G) − − EI_gain = CE['EI_macro'] - CE['EI_micro'] − </syntaxhighlight>
→代码
== 代码 ==
== 代码 ==
这里分别针对上述状态空间和变量空间的因果涌现实例，提供了计算微观、宏观有效信息的python代码，以供参考：https://github.com/KellyPPeng/effective-information-calculation
代码包括两部分：第一部分上述实例的实现，即给定微观机制和粗粒化方案，分别在状态空间和变量空间计算因果涌现；第二部分是参考https://github.com/jkbren/einet，提供的使用贪婪算法和谱分解方法，在微观转移概率矩阵中学出最优粗粒化方案，算法实现步骤拆解可参考[[复杂网络中的因果涌现]]。
== 应用 ==
== 应用 ==