Networkx - 從檢測到的社區生成的子圖的熵

Question

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我有 4 個函數用於復雜網絡分析中的一些統計計算。

import networkx as nx
import numpy as np
import math
from astropy.io import fits 

圖的度數分布：

def degree_distribution(G):
    vk = dict(G.degree())
    vk = list(vk.values()) # we get only the degree values
    maxk = np.max(vk)
    mink = np.min(min)
    kvalues= np.arange(0,maxk+1) # possible values of k
    Pk = np.zeros(maxk+1) # P(k)
    for k in vk:
        Pk[k] = Pk[k] + 1
    Pk = Pk/sum(Pk) # the sum of the elements of P(k) must to be equal to one
    
    return kvalues,Pk

圖的社區檢測：

def calculate_community_modularity(graph):
    
    communities = greedy_modularity_communities(graph) # algorithm
    modularity_dict = {} # Create a blank dictionary

    for i,c in enumerate(communities): # Loop through the list of communities, keeping track of the number for the community
        for name in c: # Loop through each neuron in a community
            modularity_dict[name] = i # Create an entry in the dictionary for the neuron, where the value is which group they belong to.

    nx.set_node_attributes(graph, modularity_dict, 'modularity')
    
    print (graph_name)
    for i,c in enumerate(communities): # Loop through the list of communities
        #if len(c) > 2: # Filter out modularity classes with 2 or fewer nodes
            print('Class '+str(i)+':', len(c)) # Print out the classes and their member numbers
    return modularity_dict
            

圖的模塊化分數：

def modularity_score(graph):
    return nx_comm.modularity(graph, nx_comm.label_propagation_communities(graph))

最后是圖熵：

def shannon_entropy(G):
    k,Pk = degree_distribution(G)
    H = 0
    for p in Pk:
        if(p > 0):
            H = H - p*math.log(p, 2)
    return H

---

問題

我現在想要實現的是找到每個社區的局部熵（變成一個子圖），並保留邊緣信息。

這可能嗎？ 怎么會這樣？

---

編輯：

正在使用的矩陣在此鏈接中：

數據集

with fits.open('mind_dataset/matrix_CEREBELLUM_large.fits') as data:
    matrix = pd.DataFrame(data[0].data.byteswap().newbyteorder())

然后將鄰接矩陣變成一個圖，“圖”或“G”，如下所示：

def matrix_to_graph(matrix):
    from_matrix = matrix.copy()
    to_numpy = from_matrix.to_numpy()
    G = nx.from_numpy_matrix(to_numpy)
    return G 

---

編輯 2 ：

根據下面提出的答案，我創建了另一個 function：

def community_entropy(modularity_dict):
    communities = {}

    #create communities as lists of nodes
    for node, community in modularity_dict.items():
        if community not in communities.keys():
            communities[community] = [node]
        else:
            communities[community].append(node)

    print(communities)
    #transform lists of nodes to actual subgraphs
    for subgraph, community in communities.items():
        communities[community] = nx.Graph.subgraph(subgraph)
        
    local_entropy = {}
    for subgraph, community in communities.items():
        local_entropy[community] = shannon_entropy(subgraph)
        
    return local_entropy

和：

cerebellum_graph = matrix_to_graph(matrix)
modularity_dict_cereb = calculate_community_modularity(cerebellum_graph)
community_entropy_cereb = community_entropy(modularity_dict_cereb)

但它會拋出錯誤：

TypeError: subgraph() missing 1 required positional argument: 'nodes'

---

有什么幫助嗎？

Answer 1

看起來，在calculate_community_modularity中，您使用greedy_modularity_communities創建了一個字典， modularity_dict ，它將您圖中的一個節點映射到一個community 。 如果我理解正確，您可以將shannon_entropy modularity_dict計算該社區的熵。

---

偽代碼

這是偽代碼，所以可能會有一些錯誤。 不過，這應該傳達原則。

運行calculate_community_modularity后，你有一個這樣的字典，其中鍵是每個節點，值是社區所屬的

modularity_dict = {node_1: community_1, node_2: community_1, node_3: community_2}

我從未使用過nx ，但看起來您可以根據節點列表提取子圖。 因此，您將遍歷您的 dict，並為每個社區創建一個節點列表。 然后您將使用該節點列表來提取該社區的實際nx子圖。

communities = {}

#create communities as lists of nodes
for node, community in modularity_dict.iteritems():
    if community not in communities.keys():
        communities[community] = [node]
    else:
        communities[community].append(node)

#transform lists of nodes to actual subgraphs
for subgraph, community in communities.iteritems():
    communities[community] = networkx.Graph.subgraph(subgraph)

既然communities是一個帶有社區 id 鍵的字典，以及定義該社區的nx子圖的值，您應該能夠通過shannon_entropy運行這些子圖，因為子圖的類型與您的類型相同原始圖

local_entropy = {}
for subgraph, community in communities.iteritems():
    local_entropy[community] = shannon_entropy(subgraph)

Answer 2

使用我在此處提供的代碼作為您的問題的答案，從社區創建圖表。 您可以首先為每個社區創建不同的圖表（基於圖表的社區邊緣屬性）。 然后，您可以使用您的shannon_entropy和degree_distribution function 計算每個社區的熵。

根據您在上面提到的其他問題中提供的空手道俱樂部示例，請參閱下面的代碼：

import networkx as nx
import networkx.algorithms.community as nx_comm
import matplotlib.pyplot as plt
import numpy as np
import math

def degree_distribution(G):
    vk = dict(G.degree())
    vk = list(vk.values()) # we get only the degree values
    maxk = np.max(vk)
    mink = np.min(min)
    kvalues= np.arange(0,maxk+1) # possible values of k
    Pk = np.zeros(maxk+1) # P(k)
    for k in vk:
        Pk[k] = Pk[k] + 1
    Pk = Pk/sum(Pk) # the sum of the elements of P(k) must to be equal to one
    
    return kvalues,Pk

def shannon_entropy(G):
    k,Pk = degree_distribution(G)
    H = 0
    for p in Pk:
        if(p > 0):
            H = H - p*math.log(p, 2)
    return H


G = nx.karate_club_graph()

# Find the communities
communities = sorted(nx_comm.greedy_modularity_communities(G), key=len, reverse=True)

# Count the communities
print(f"The club has {len(communities)} communities.")

'''Add community to node attributes'''
for c, v_c in enumerate(communities):
    for v in v_c:
        # Add 1 to save 0 for external edges
        G.nodes[v]['community'] = c + 1

'''Find internal edges and add their community to their attributes'''
for v, w, in G.edges:
    if G.nodes[v]['community'] == G.nodes[w]['community']:
        # Internal edge, mark with community
        G.edges[v, w]['community'] = G.nodes[v]['community']
    else:
        # External edge, mark as 0
        G.edges[v, w]['community'] = 0


N_coms=len(communities)
edges_coms=[]#edge list for each community
coms_G=[nx.Graph() for _ in range(N_coms)] #community graphs
colors=['tab:blue','tab:orange','tab:green']
fig=plt.figure(figsize=(12,5))

for i in range(N_coms):
  edges_coms.append([(u,v,d) for u,v,d in G.edges(data=True) if d['community'] == i+1])#identify edges of interest using the edge attribute
  coms_G[i].add_edges_from(edges_coms[i]) #add edges

ent_coms=[shannon_entropy(coms_G[i]) for i in range(N_coms)] #Compute entropy
for i in range(N_coms):
  plt.subplot(1,3,i+1)#plot communities
  plt.title('Community '+str(i+1)+ ', entropy: '+str(np.round(ent_coms[i],1)))
  pos=nx.circular_layout(coms_G[i])
  nx.draw(coms_G[i],pos=pos,with_labels=True,node_color=colors[i])  

output 給出：