# 导入库
import networkx as nx
import matplotlib.pyplot as plt
K-近邻规则网络的生成与可视化
def regular_graph(n, k):
G = nx.Graph()
nodes = list(range(n)) # 节点标签设置为0到n-1
# 每个节点与周围k/2个邻近节点相连
for j in range(1, k // 2 + 1): # //是取整除 - 返回商的整数部分（向下取整）
targets = nodes[j:] + nodes[0:j]
G.add_edges_from(zip(nodes, targets))
return G
n = 20 # 网络节点总数
k = 4 # 近邻节点数
color_list = ['red','gray']
G = regular_graph(n, k)
pos = nx.circular_layout(G)
nx.draw(G, pos, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("K-regular", fontsize=20)
WS小世界网络的生成与可视化
plt.figure(figsize=(15, 4))
# 绘制规则网络
p = 0
G1 = nx.watts_strogatz_graph(n, k, p)
plt.subplot(131)
pos1 = nx.circular_layout(G1)
nx.draw(G1, pos1, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("regular", fontsize=20)
# 绘制WS小世界
p=0.2
G2 = nx.watts_strogatz_graph(n, k, p)
plt.subplot(132)
pos2 = nx.circular_layout(G2)
nx.draw(G2, pos2, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("small-world", fontsize=20)
#绘制随机网络
p=1.0
G3 = nx.watts_strogatz_graph(n, k, p)
plt.subplot(133)
pos3 = nx.circular_layout(G3)
nx.draw(G3, pos3, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("random", fontsize=20)
NW小世界网络的生成与可视化
plt.figure(figsize=(15, 4))
# 绘制规则网络
p = 0
G1 = nx.watts_strogatz_graph(n, k, p)
plt.subplot(131)
pos1 = nx.circular_layout(G1)
nx.draw(G1, pos1, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("regular", fontsize=20)
# 绘制NW小世界
p=0.2
G2 = nx.newman_watts_strogatz_graph(n, k, p)
plt.subplot(132)
pos2 = nx.circular_layout(G2)
nx.draw(G2, pos2, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("small-world", fontsize=20)
#绘制完全网络：加边概率为1
G3 = nx.complete_graph(n)
plt.subplot(133)
pos3 = nx.circular_layout(G3)
nx.draw(G3, pos3, node_size=100, node_color=color_list[0], edge_color=color_list[1])
plt.title("complete", fontsize=20)
plt.savefig("NW.png", dpi=600)
WS小世界网络的度分布
import numpy as np
# 定义求度分布的函数
def get_pdf(G, k):
N = len(G.nodes())
Pk = []
for ki in k:
c = 0
for i in G.nodes():
if G.degree(i) == ki:
c += 1
Pk.append(c/N)
return Pk
# 以N＝1000，K＝6的WS模型的数值模拟结果为例
N = 1000
K = 6
samples = 100 # 统计平均次数
p_rew = [0.1,0.2,0.4,0.6,1.0]
plt.figure(figsize=(8,6))
symbols = ["ro-","bs-","g*-","yv-","k^-"]
# 为了便于统计平均，指定区间[1,16]
kmin, kmax = 1, 16
x = list(range(kmin, kmax+1))
c = 0
for p in p_rew:
s = np.zeros(kmax-kmin+1)
for i in range(samples):
G = nx.watts_strogatz_graph(N, K, p)
y = get_pdf(G, x)
s += np.array(y)
s = list(s)
# 剔除概率为零的点：这里也可以用numpy的函数直接剔除更方便一些
new_x = []
new_y = []
for i in range(len(x)):
if s[i] != 0:
new_x.append(x[i])
new_y.append(s[i])
plt.plot(new_x, np.array(new_y)/samples, symbols[c], label='$p_{rew} = $'+str(p))
c += 1
plt.legend(loc=0, fontsize=14)
plt.xlabel("$k$")
plt.ylabel("$p_k$")
plt.yscale("log")
plt.xlim([kmin, kmax])
WS小世界网络的“小世界”与“高集聚”特性
# 设置初始参数
N, K = 1000, 10
samples = 10
p_rew = np.logspace(0,4,10)/10000
p_rew
# 平均距离与平均集聚系数
C = []
CT = [] # 理论近似值：{[3(K-2)]/[4(K-1)]}*(1-p)^3
L = []
for p in p_rew:
s1 = 0
s2 = 0
for i in range(samples):
G = nx.watts_strogatz_graph(N, K, p)
# 为了防止在计算平均距离时报错：最好改用生成连通WS小世界网络函数connected_watts_strogatz_graph()
s1 += nx.average_clustering(G)
s2 += nx.average_shortest_path_length(G)
ct = (3*(K-2)/(4*(K-1)))*((1-p)**3)
CT.append(ct)
C.append(s1/samples)
L.append(s2/samples)
plt.figure(figsize=(10,4))
plt.subplot(121)
plt.plot(p_rew, C, 'ro', label='$C(p)$')
plt.plot(p_rew, CT, 'r-', label='$CT(p)$')
plt.legend(loc=0, fontsize=14)
plt.xlabel("$p$")
plt.xscale("log")
plt.subplot(122)
plt.plot(p_rew, L, 'bs', label='$L(p)$')
plt.legend(loc=0, fontsize=14)
plt.xlabel("$p$")
plt.xscale("log")
plt.yscale("log")
# 初始规则网络的平均集聚系数和平均距离
G0 = nx.watts_strogatz_graph(N, K, 0)
C0 = nx.average_clustering(G0)
L0 = nx.average_shortest_path_length(G0)
plt.figure(figsize=(6,4))
plt.plot(p_rew, np.array(C)/C0, 'ro', label='$C(p)/C(0)$')
plt.plot(p_rew, np.array(L)/L0, 'bs', label='$L(p)/L(0)$')
plt.legend(loc=0, fontsize=14)
plt.xlabel("$p$")
plt.xscale("log")
真实网络的集聚系数与度之间的依赖关系（以科学合作网络为例）
def C_vs_k(G):
klist = [G.degree(i) for i in G.nodes()]
# 计算每个节点的集聚系数
all_C = {i: nx.clustering(G, i) for i in G.nodes()}
all_k = list(set(klist)) # 所有可能的度值
# 计算度值为k的节点的集聚系数的平均值
C_k = {}
for k in sorted(all_k):
s = 0
j = 0
for i in G.nodes():
if G.degree(i) == k:
j = j + 1
s = s + all_C[i]
avc_k = s/j
C_k[k] = avc_k
return C_k
import pandas as pd
df = pd.read_csv("citation.csv")
G = nx.from_pandas_edgelist(df, 'source', 'target', create_using = nx.Graph())
len(G.nodes())
C_k = C_vs_k(G)
avC = nx.average_clustering(G)
print(avC)
x = np.linspace(1,10000,10000)
y = [avC]*10000
plt.figure(figsize=(6,4))
plt.plot(C_k.keys(), C_k.values(), 'ro')
plt.plot(x, y, 'b-')
plt.xlabel("$k$")
plt.ylabel("$C(k)$")
plt.xscale("log")
plt.yscale("log")
plt.xlim([1,1e4])