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我来详细介绍如何使用NetworkX进行同配性分析（Assortativity Analysis）。
基础同配性分析
节点度同配性
import networkx as nx
import numpy as np
import matplotlib.pyplot as plt
# 创建示例图
G = nx.karate_club_graph() # 使用空手道俱乐部图
# 计算度同配性系数
assortativity_degree = nx.degree_assortativity_coefficient(G)
print(f"度同配性系数: {assortativity_degree:.4f}")
# 解释：
# 正值：高度节点倾向于连接高度节点（同配）
# 负值：高度节点倾向于连接低度节点（异配）
# 接近0：无明显模式
多重属性同配性
# 计算其他属性的同配性
# 1. 数值属性同配性
G = nx.karate_club_graph()
# 添加数值属性
for node in G.nodes():
G.nodes[node]['age'] = np.random.randint(20, 60)
assortativity_age = nx.numeric_assortativity_coefficient(G, 'age')
print(f"年龄同配性: {assortativity_age:.4f}")
# 2. 类别属性同配性
# 添加类别属性
for node in G.nodes():
G.nodes[node]['club'] = 'Officer' if node < 20 else 'Mr. Hi'
assortativity_club = nx.attribute_assortativity_coefficient(G, 'club')
print(f"俱乐部同配性: {assortativity_club:.4f}")
高级同配性分析
混合矩阵分析
import pandas as pd
def analyze_assortativity_matrix(G, attribute):
"""分析同配性混合矩阵"""
# 获取混合矩阵
M = nx.attribute_mixing_matrix(G, attribute)
# 创建可读的DataFrame
nodes_attr = nx.get_node_attributes(G, attribute)
unique_attrs = sorted(set(nodes_attr.values()))
df = pd.DataFrame(M, index=unique_attrs, columns=unique_attrs)
print("混合矩阵:")
print(df)
return df
# 使用示例
G = nx.karate_club_graph()
# 添加社区标签作为属性
communities = nx.community.greedy_modularity_communities(G)
for i, comm in enumerate(communities):
for node in comm:
G.nodes[node]['community'] = f'Community_{i}'
matrix_df = analyze_assortativity_matrix(G, 'community')
度-度相关性分析
def analyze_degree_correlation(G):
"""分析度-度相关性"""
# 计算每个节点对的度
degrees = dict(G.degree())
# 收集边两端的度
degree_pairs = []
for u, v in G.edges():
degree_pairs.append((degrees[u], degrees[v]))
# 转换为numpy数组
degree_pairs = np.array(degree_pairs)
# 计算平均邻居度
avg_neighbor_degree = nx.average_neighbor_degree(G)
return degree_pairs, avg_neighbor_degree
# 使用示例
degree_pairs, avg_neighbor_degree = analyze_degree_correlation(G)
# 可视化度-度相关性
plt.figure(figsize=(12, 5))
plt.subplot(1, 2, 1)
plt.scatter(degree_pairs[:, 0], degree_pairs[:, 1], alpha=0.5)
plt.xlabel('节点度')
plt.ylabel('邻居节点度')'度-度相关性散点图')
plt.plot([0, max(degree_pairs.flatten())],
[0, max(degree_pairs.flatten())], 'r--', alpha=0.5)
plt.subplot(1, 2, 2)
degrees_list = sorted(avg_neighbor_degree.keys())
knn_values = [avg_neighbor_degree[k] for k in degrees_list]
plt.plot(degrees_list, knn_values, 'o-')
plt.xlabel('节点度 k')
plt.ylabel('平均邻居度 <knn>')'平均邻居度 vs 节点度')
plt.tight_layout()
plt.show()
实际应用案例
社交网络分析
def social_network_assortativity_analysis(G):
"""社交网络同配性综合分析"""
results = {}
# 1. 度同配性
results['degree_assortativity'] = nx.degree_assortativity_coefficient(G)
# 2. 如果有节点属性，分析属性同配性
node_attributes = {}
# 假设有性别属性
if 'gender' in nx.get_node_attributes(G, 'gender'):
results['gender_assortativity'] = nx.attribute_assortativity_coefficient(G, 'gender')
# 假设有年龄属性
if 'age' in nx.get_node_attributes(G, 'age'):
results['age_assortativity'] = nx.numeric_assortativity_coefficient(G, 'age')
# 3. 计算局部同配性
results['local_assortativity'] = local_assortativity_analysis(G)
return results
def local_assortativity_analysis(G):
"""局部同配性分析"""
local_assort = {}
degrees = dict(G.degree())
for node in G.nodes():
neighbors = list(G.neighbors(node))
if len(neighbors) > 1:
neighbor_degrees = [degrees[n] for n in neighbors]
# 计算局部同配性
local_assort[node] = np.corrcoef(
[degrees[node]] * len(neighbor_degrees),
neighbor_degrees
)[0, 1]
else:
local_assort[node] = np.nan
return local_assort
网络演化中的同配性
def temporal_assortativity_analysis(graphs_list):
"""时间序列同配性分析"""
time_points = len(graphs_list)
assortativity_over_time = []
for t, G in enumerate(graphs_list):
if G.number_of_edges() > 0:
assort = nx.degree_assortativity_coefficient(G)
assortativity_over_time.append(assort)
else:
assortativity_over_time.append(np.nan)
return assortativity_over_time
# 模拟时间序列
def create_temporal_graphs():
"""创建随时间变化的图序列"""
graphs = []
base_G = nx.karate_club_graph()
for t in range(5):
G = base_G.copy()
# 随机添加一些边来模拟演化
for _ in range(10):
u, v = np.random.choice(G.nodes(), 2, replace=False)
if not G.has_edge(u, v):
G.add_edge(u, v)
graphs.append(G)
return graphs
# 使用示例
temporal_graphs = create_temporal_graphs()
assort_over_time = temporal_assortativity_analysis(temporal_graphs)
plt.figure(figsize=(10, 6))
plt.plot(range(len(assort_over_time)), assort_over_time, 'o-')
plt.xlabel('时间步')
plt.ylabel('度同配性系数')'网络同配性随时间变化')
plt.grid(True)
plt.show()
可视化同配性
def visualize_assortativity(G):
"""可视化网络同配性"""
fig, axes = plt.subplots(2, 2, figsize=(15, 12))
# 1. 原始网络
pos = nx.spring_layout(G, seed=42)
nx.draw(G, pos, ax=axes[0, 0], node_color='lightblue',
node_size=100, with_labels=True)
axes[0, 0].set_title('原始网络')
# 2. 按度着色的网络
degrees = dict(G.degree())
node_colors = [degrees[n] for n in G.nodes()]
nx.draw(G, pos, ax=axes[0, 1], node_color=node_colors,
node_size=100, with_labels=True,
cmap=plt.cm.RdYlBu)
axes[0, 1].set_title('节点度分布')
# 3. 同配性矩阵热图
from networkx.algorithms.assortativity import mixing_matrix
mixing_mat = mixing_matrix(G)
im = axes[1, 0].imshow(mixing_mat, cmap='viridis', aspect='auto')
axes[1, 0].set_title('度混合矩阵')
plt.colorbar(im, ax=axes[1, 0])
# 4. 平均邻居度图
knn = nx.average_neighbor_degree(G)
degrees_list = list(knn.keys())
knn_values = list(knn.values())
axes[1, 1].scatter(degrees_list, knn_values)
axes[1, 1].set_xlabel('节点度 k')
axes[1, 1].set_ylabel('平均邻居度 <knn>')
axes[1, 1].set_title('knn(k) 曲线')
axes[1, 1].grid(True)
plt.tight_layout()
plt.show()
# 使用示例
G = nx.karate_club_graph()
visualize_assortativity(G)
统计显著性检验
def assortativity_significance_test(G, n_permutations=1000):
"""同配性显著性检验"""
original_assortativity = nx.degree_assortativity_coefficient(G)
permutations = []
for _ in range(n_permutations):
# 随机重连边（保持度序列不变）
G_perm = nx.double_edge_swap(G, nswap=len(G.edges())*2)
if G_perm:
perm_assort = nx.degree_assortativity_coefficient(G_perm)
permutations.append(perm_assort)
# 计算p值
permutations = np.array(permutations)
p_value = np.mean(np.abs(permutations) >= np.abs(original_assortativity))
# 计算z-score
z_score = (original_assortativity - np.mean(permutations)) / np.std(permutations)
results = {
'original': original_assortativity,
'mean_null': np.mean(permutations),
'std_null': np.std(permutations),
'z_score': z_score,
'p_value': p_value
}
return results
# 使用示例
G = nx.karate_club_graph()
significance_results = assortativity_significance_test(G)
print("同配性显著性检验结果:")
for key, value in significance_results.items():
print(f"{key}: {value:.4f}")
这些代码示例涵盖了NetworkX中同配性分析的主要方面,根据你的具体需求，可以选择使用基础分析、高级分析或可视化方法。
