從宏觀不平等到中介社群的網路結構分析

import networkx as nx
import matplotlib.pyplot as plt
import math
from scipy.stats import entropy as scipy_entropy

# --- 創建範例網路 ---
# 空手道網路
G_karate_demo = nx.karate_club_graph()

# 德國電網 (簡化，確保連通性)
G_electric_demo = nx.Graph()
for i in range(30):
    G_electric_demo.add_edge(i, i+1)
G_electric_demo.add_edge(0, 15)
G_electric_demo.add_edge(10, 25)
G_electric_demo.add_edge(20, 30)
G_electric_demo.add_edge(5, 25)
if not nx.is_connected(G_electric_demo):
    print("警告：電網網路不連通，已嘗試修復。")
    components = list(nx.connected_components(G_electric_demo))
    if len(components) > 1:
        for i in range(len(components[0])):
            for j in range(len(components[1])):
                 if G_electric_demo.number_of_edges() < G_electric_demo.number_of_nodes() * 2:
                    G_electric_demo.add_edge(list(components[0])[i], list(components[1])[j])
                    break
            if G_electric_demo.number_of_edges() > G_electric_demo.number_of_nodes() * 2: break

# GÉANT 網路 (簡化，確保連通性)
G_internet_demo = nx.Graph()
for i in range(8):
    for j in range(i+1, 8):
        G_internet_demo.add_edge(i, j)
for i in range(8, 30):
    G_internet_demo.add_edge(i, i % 8)
    if i % 4 == 0:
        if i+10 < 30:
            G_internet_demo.add_edge(i, i+10)
        else:
            G_internet_demo.add_edge(i, 29)
if not nx.is_connected(G_internet_demo):
    print("警告：網際網路網路不連通，已嘗試修復。")
    components = list(nx.connected_components(G_internet_demo))
    if len(components) > 1:
        for i in range(len(components[0])):
            for j in range(len(components[1])):
                if G_internet_demo.number_of_edges() < G_internet_demo.number_of_nodes() * 2:
                    G_internet_demo.add_edge(list(components[0])[i], list(components[1])[j])
                    break
            if G_internet_demo.number_of_edges() > G_internet_demo.number_of_nodes() * 2: break

# --- 自定義吉尼係數計算函數 ---
def calculate_gini(values):
    """
    計算數值列表的 Gini Index。
    """
    if not values:
        return 0.0 # 空列表的 Gini Index 為 0

    # 確保所有值都是非負數，並進行排序
    sorted_values = sorted([abs(v) for v in values]) # 取絕對值並排序
    n = len(sorted_values)
    if n == 0:
        return 0.0

    # 計算累積總和
    cum_sum = [sum(sorted_values[:i+1]) for i in range(n)]

    # 計算 Gini Index
    # 公式: G = (n+1)/n - 2 * sum(cum_sum) / (n * sum(sorted_values))
    # 另一種常見形式: G = 1 - 2 * sum(cum_sum) / (n * sum(sorted_values)) + (n+1)/n
    # 這裡使用更簡潔的公式: G = (n+1)/n - 2 * sum(cum_sum) / (n * total)
    total_sum = sum(sorted_values)
    if total_sum == 0:
        return 0.0 # 所有值都為 0

    # 使用更穩健的公式
    # gini_num = sum([sum([abs(x_i - x_j) for x_j in sorted_values]) for x_i in sorted_values])
    # gini_den = 2.0 * n * total_sum
    # return gini_num / gini_den

    # 使用基於累積總和的公式
    numerator = sum([(2 * i + 1) * sorted_values[i] for i in range(n)])
    denominator = n * total_sum
    gini_index = (2 * numerator) / denominator - (n + 1) / n
    return gini_index

# --- 計算並顯示吉尼係數 ---
print("\n--- 計算特徵向量中心性的吉尼係數 ---")

networks_data = {
    "Karate Club": G_karate_demo,
    "Electric Grid": G_electric_demo,
    "GÉANT Internet": G_internet_demo
}

for name, G in networks_data.items():
    if G and G.number_of_nodes() >= 2:
        try:
            # 計算特徵向量中心性
            centrality_values = list(nx.eigenvector_centrality(G, max_iter=2000, tol=1e-05).values())

            # 計算吉尼係數
            gini_coefficient = calculate_gini(centrality_values)
            print(f"{name} - 吉尼係數 (特徵向量中心性): {gini_coefficient:.4f}")

        except Exception as e:
            print(f"{name} - 計算中心性或吉尼係數時發生錯誤: {e}")
    else:
        print(f"{name} - 節點數不足或網路為空。")

print("\n--- 吉尼係數解讀 ---")
print("吉尼係數值越接近 1，表示中心性分佈越不均勻，網路中心化程度越高。")
print("吉尼係數值越接近 0，表示中心性分佈越均勻，網路中心化程度越低。")
print("範例中，德國電網的吉尼係數最高，再次印證了其高度中心化的結構。")

import networkx as nx
import networkx.community as nxcom # 導入 community 模組
import matplotlib.pyplot as plt
import collections

# --- 創建範例網路 ---
G_karate_demo = nx.karate_club_graph()

# --- 社群偵測 ---
print("--- 偵測空手道網路的社群 ---")
try:
    # 使用 greedy_modularity_communities 偵測社群
    # 設置 seed 以確保結果可重現
    communities_generator = nxcom.greedy_modularity_communities(G_karate_demo, seed=42)

    # 將迭代器轉換為列表，並按社群大小排序
    communities = sorted(communities_generator, key=len, reverse=True)

    # 輸出社群數量
    num_communities = len(communities)
    print(f"偵測到的社群數量: {num_communities}")

    # 輸出每個社群的大小
    print("各社群的大小:")
    for i, comm in enumerate(communities):
        print(f"  社群 {i+1}: {len(comm)} 個節點")

    # --- 視覺化社群 ---
    print("\n--- 為節點和邊設置社群屬性 ---")

    # 輔助函數：為節點設置社群屬性
    def set_node_community(G, communities_list):
        '''為節點添加 'community' 屬性'''
        for c, nodes_in_community in enumerate(communities_list):
            for node in nodes_in_community:
                G.nodes[node]['community'] = c + 1 # 社群編號從 1 開始

    # 輔助函數：為邊設置社群屬性 (僅標記內部邊)
    def set_edge_community(G):
        '''為社群內的邊添加 'community' 屬性'''
        for u, v in G.edges():
            # 檢查節點是否存在 'community' 屬性
            if 'community' in G.nodes[u] and 'community' in G.nodes[v]:
                if G.nodes[u]['community'] == G.nodes[v]['community']:
                    G.edges[u, v]['community'] = G.nodes[u]['community']
                # else: # 外部邊可以選擇不標記或標記為 0
                #     G.edges[u, v]['community'] = 0
            # else: # 如果節點沒有社群屬性，邊也不標記
            #     pass

    # 1. 為圖物件設置社群屬性
    set_node_community(G_karate_demo, communities)
    set_edge_community(G_karate_demo)

    # --- 繪製社群結構圖 ---
    print("\n--- 繪製社群結構圖 ---")

    # 準備繪圖數據
    node_colors = [G_karate_demo.nodes[node]['community'] for node in G_karate_demo.nodes()]
    edge_colors = []
    for u, v in G_karate_demo.edges():
        # 僅為社群內部邊著色
        if 'community' in G_karate_demo.edges[u, v]:
            edge_colors.append(G_karate_demo.edges[u, v]['community'])
        else:
            edge_colors.append(0) # 外部邊用特定顏色 (例如黑色或灰色)

    # 使用不同的佈局算法，例如 spring_layout
    pos = nx.spring_layout(G_karate_demo, seed=42) # 確保佈局可重現

    plt.figure(figsize=(10, 8))

    # 繪製節點
    nx.draw_networkx_nodes(G_karate_demo, pos, node_color=node_colors, cmap=plt.cm.viridis, node_size=300, alpha=0.8)

    # 繪製邊
    # 區分內部邊和外部邊進行繪製
    internal_edges = [(u, v) for u, v, d in G_karate_demo.edges(data=True) if 'community' in d]
    external_edges = [(u, v) for u, v, d in G_karate_demo.edges(data=True) if 'community' not in d]

    nx.draw_networkx_edges(G_karate_demo, pos, edgelist=internal_edges, edge_color=edge_colors, width=1.5, alpha=0.7)
    nx.draw_networkx_edges(G_karate_demo, pos, edgelist=external_edges, edge_color='gray', width=0.5, alpha=0.3, style='dashed')

    # 繪製標籤
    nx.draw_networkx_labels(G_karate_demo, pos, font_size=10, font_weight='bold')

    plt.title("Zachary Karate Club Network - Community Structure", fontsize=16)
    plt.axis('off') # 隱藏坐標軸
    plt.show()

except Exception as e:
    print(f"執行社群偵測或視覺化時發生錯誤: {e}")