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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 网络科学理论
网络科学：使用NetworkX分析复杂网络
王成军 
wangchengjun@nju.edu.cn
计算传播网 http
Step1:  WWW Data download
http
Step2:  描述网络
nx.karate_club_graph
我们从karate_club_graph开始，探索网络的基本性质。
Step3:  网络直径
Step4:  密度
Step5:  作业：
计算www网络的网络密度
聚集系数
Step6:  Spacing in Math Mode
In a math environment, LaTeX ignores the spaces you type and puts in the spacing that it thinks is best. LaTeX formats mathematics the way it's done in mathematics texts. If you want different spacing, LaTeX provides the following four commands for use in math mode
Step7:  Degree centrality measures.（度中心性）
degree_centrality(G)        # Compute the degree centrality for nodes.
in_degree_centrality(G)     # Compute the in-degree centrality for nodes.
out_degree_centrality(G)    # Compute the out-degree centrality for nodes.
closeness_centrality(G[, v, weighted_edges])   #  Compute closeness centrality for nodes.
betweenness_centrality(G[, normalized, ...])  #  Betweenness centrality measures.（介数中心性）
Step8:  度分布
Step9:  网络科学理论简介
网络科学：分析网络结构
王成军 
wangchengjun@nju.edu.cn
计算传播网 http
Step10:  ER随机网络
Step11:  小世界网络
Step12:  BA网络
Step13:  作业：
阅读 Barabasi (1999) Internet Diameter of the world wide web.Nature.401
绘制www网络的出度分布、入度分布
使用BA模型生成节点数为N、幂指数为$\gamma$的网络
计算平均路径长度d与节点数量的关系
<img src = './img/diameter.png' width = 10000>
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Python Code:
%matplotlib inline
import networkx as nx
import matplotlib.cm as cm
import matplotlib.pyplot as plt
import networkx as nx
G=nx.Graph() # G = nx.DiGraph() # 有向网络
# 添加（孤立）节点
G.add_node("spam")
# 添加节点和链接
G.add_edge(1,2)
print(G.nodes())
print(G.edges())
# 绘制网络
nx.draw(G, with_labels = True)
Explanation: 网络科学理论
网络科学：使用NetworkX分析复杂网络
王成军 
wangchengjun@nju.edu.cn
计算传播网 http://computational-communication.com
http://networkx.readthedocs.org/en/networkx-1.11/tutorial/
End of explanation
G = nx.Graph()
n = 0
with open ('/Users/chengjun/bigdata/www.dat.gz.txt') as f:
    for line in f:
        n += 1
        #if n % 10**4 == 0:
            #flushPrint(n)
        x, y = line.rstrip().split(' ')
        G.add_edge(x,y)
nx.info(G)
Explanation: WWW Data download
http://www3.nd.edu/~networks/resources.htm
World-Wide-Web: [README] [DATA]
Réka Albert, Hawoong Jeong and Albert-László Barabási:
Diameter of the World Wide Web Nature 401, 130 (1999) [ PDF ]
作业：
下载www数据
构建networkx的网络对象g（提示：有向网络）
将www数据添加到g当中
计算网络中的节点数量和链接数量
End of explanation
G = nx.karate_club_graph()
 
clubs = [G.node[i]['club'] for i in G.nodes()]
colors = []
for j in clubs:
    if j == 'Mr. Hi':
        colors.append('r')
    else:
        colors.append('g')
 
nx.draw(G,  with_labels = True, node_color = colors)
G.node[1] # 节点1的属性
G.edge.keys()[:3] # 前三条边的id
nx.info(G)
G.nodes()[:10]
G.edges()[:3]
G.neighbors(1)
nx.average_shortest_path_length(G) 
Explanation: 描述网络
nx.karate_club_graph
我们从karate_club_graph开始，探索网络的基本性质。
End of explanation
nx.diameter(G)#返回图G的直径（最长最短路径的长度）
Explanation: 网络直径
End of explanation
nx.density(G)
nodeNum = len(G.nodes())
edgeNum = len(G.edges())
2.0*edgeNum/(nodeNum * (nodeNum - 1))
Explanation: 密度
End of explanation
cc = nx.clustering(G)
cc.items()[:5]
plt.hist(cc.values(), bins = 15)
plt.xlabel('$Clustering \, Coefficient, \, C$', fontsize = 20)
plt.ylabel('$Frequency, \, F$', fontsize = 20)
plt.show()
Explanation: 作业：
计算www网络的网络密度
聚集系数
End of explanation
# M. E. J. Newman, Mixing patterns in networks Physical Review E, 67 026126, 2003
nx.degree_assortativity_coefficient(G) #计算一个图的度匹配性。
Ge=nx.Graph()
Ge.add_nodes_from([0,1],size=2)
Ge.add_nodes_from([2,3],size=3)
Ge.add_edges_from([(0,1),(2,3)])
print(nx.numeric_assortativity_coefficient(Ge,'size'))
# plot degree correlation  
from collections import defaultdict
import numpy as np
l=defaultdict(list)
g = nx.karate_club_graph()
for i in g.nodes():
    k = []
    for j in g.neighbors(i):
        k.append(g.degree(j))
    l[g.degree(i)].append(np.mean(k))   
    #l.append([g.degree(i),np.mean(k)])
  
x = l.keys()
y = [np.mean(i) for i in l.values()]
#x, y = np.array(l).T
plt.plot(x, y, 'r-o', label = '$Karate\;Club$')
plt.legend(loc=1,fontsize=10, numpoints=1)
plt.xscale('log'); plt.yscale('log')
plt.ylabel(r'$<knn(k)$> ', fontsize = 20)
plt.xlabel('$k$', fontsize = 20)
plt.show()
Explanation: Spacing in Math Mode
In a math environment, LaTeX ignores the spaces you type and puts in the spacing that it thinks is best. LaTeX formats mathematics the way it's done in mathematics texts. If you want different spacing, LaTeX provides the following four commands for use in math mode:
\; - a thick space
\: - a medium space
\, - a thin space
\! - a negative thin space
匹配系数
End of explanation
dc = nx.degree_centrality(G)
closeness = nx.closeness_centrality(G)
betweenness= nx.betweenness_centrality(G)
fig = plt.figure(figsize=(15, 4),facecolor='white')
ax = plt.subplot(1, 3, 1)
plt.hist(dc.values(), bins = 20)
plt.xlabel('$Degree \, Centrality$', fontsize = 20)
plt.ylabel('$Frequency, \, F$', fontsize = 20)
ax = plt.subplot(1, 3, 2)
plt.hist(closeness.values(), bins = 20)
plt.xlabel('$Closeness \, Centrality$', fontsize = 20)
ax = plt.subplot(1, 3, 3)
plt.hist(betweenness.values(), bins = 20)
plt.xlabel('$Betweenness \, Centrality$', fontsize = 20)
plt.tight_layout()
plt.show()
fig = plt.figure(figsize=(15, 8),facecolor='white')
for k in betweenness:
    plt.scatter(dc[k], closeness[k], s = betweenness[k]*1000)
    plt.text(dc[k], closeness[k]+0.02, str(k))
plt.xlabel('$Degree \, Centrality$', fontsize = 20)
plt.ylabel('$Closeness \, Centrality$', fontsize = 20)
plt.show()
Explanation: Degree centrality measures.（度中心性）
degree_centrality(G)        # Compute the degree centrality for nodes.
in_degree_centrality(G)     # Compute the in-degree centrality for nodes.
out_degree_centrality(G)    # Compute the out-degree centrality for nodes.
closeness_centrality(G[, v, weighted_edges])   #  Compute closeness centrality for nodes.
betweenness_centrality(G[, normalized, ...])  #  Betweenness centrality measures.（介数中心性）
End of explanation
from collections import defaultdict
import numpy as np
def plotDegreeDistribution(G):
    degs = defaultdict(int)
    for i in G.degree().values(): degs[i]+=1
    items = sorted ( degs.items () )
    x, y = np.array(items).T
    y_sum = np.sum(y)
    y = [float(i)/y_sum for i in y]
    plt.plot(x, y, 'b-o')
    plt.xscale('log')
    plt.yscale('log')
    plt.legend(['Degree'])
    plt.xlabel('$K$', fontsize = 20)
    plt.ylabel('$P_K$', fontsize = 20)
    plt.title('$Degree\,Distribution$', fontsize = 20)
    plt.show()   
    
G = nx.karate_club_graph()     
plotDegreeDistribution(G)
Explanation: 度分布
End of explanation
import networkx as nx
import matplotlib.pyplot as plt
RG = nx.random_graphs.random_regular_graph(3,200)  #生成包含200个节点、每个节点有3个邻居的规则图RG
pos = nx.spectral_layout(RG)          #定义一个布局，此处采用了spectral布局方式，后变还会介绍其它布局方式，注意图形上的区别
nx.draw(RG,pos,with_labels=False,node_size = 30)  #绘制规则图的图形，with_labels决定节点是非带标签（编号），node_size是节点的直径
plt.show()  #显示图形
plotDegreeDistribution(RG)
Explanation: 网络科学理论简介
网络科学：分析网络结构
王成军 
wangchengjun@nju.edu.cn
计算传播网 http://computational-communication.com
规则网络
End of explanation
import networkx as nx
import matplotlib.pyplot as plt
ER = nx.random_graphs.erdos_renyi_graph(200,0.05)  #生成包含20个节点、以概率0.2连接的随机图
pos = nx.shell_layout(ER)          #定义一个布局，此处采用了shell布局方式
nx.draw(ER,pos,with_labels=False,node_size = 30) 
plt.show()
plotDegreeDistribution(ER)
Explanation: ER随机网络
End of explanation
import networkx as nx
import matplotlib.pyplot as plt
WS = nx.random_graphs.watts_strogatz_graph(200,4,0.3)  #生成包含200个节点、每个节点4个近邻、随机化重连概率为0.3的小世界网络
pos = nx.circular_layout(WS)          #定义一个布局，此处采用了circular布局方式
nx.draw(WS,pos,with_labels=False,node_size = 30)  #绘制图形
plt.show()
plotDegreeDistribution(WS)
nx.diameter(WS)
cc = nx.clustering(WS)
plt.hist(cc.values(), bins = 10)
plt.xlabel('$Clustering \, Coefficient, \, C$', fontsize = 20)
plt.ylabel('$Frequency, \, F$', fontsize = 20)
plt.show()
import numpy as np
np.mean(cc.values())
Explanation: 小世界网络
End of explanation
import networkx as nx
import matplotlib.pyplot as plt
BA= nx.random_graphs.barabasi_albert_graph(200,2)  #生成n=20、m=1的BA无标度网络
pos = nx.spring_layout(BA)          #定义一个布局，此处采用了spring布局方式
nx.draw(BA,pos,with_labels=False,node_size = 30)  #绘制图形
plt.show()
plotDegreeDistribution(BA)
BA= nx.random_graphs.barabasi_albert_graph(20000,2)  #生成n=20、m=1的BA无标度网络
plotDegreeDistribution(BA)
import networkx as nx
import matplotlib.pyplot as plt
BA= nx.random_graphs.barabasi_albert_graph(500,1)  #生成n=20、m=1的BA无标度网络
pos = nx.spring_layout(BA)          #定义一个布局，此处采用了spring布局方式
nx.draw(BA,pos,with_labels=False,node_size = 30)  #绘制图形
plt.show()
nx.degree_histogram(BA)[:3]
BA.degree().items()[:3]
plt.hist(BA.degree().values())
plt.show()
from collections import defaultdict
import numpy as np
def plotDegreeDistributionLongTail(G):
    degs = defaultdict(int)
    for i in G.degree().values(): degs[i]+=1
    items = sorted ( degs.items () )
    x, y = np.array(items).T
    y_sum = np.sum(y)
    y = [float(i)/y_sum for i in y]
    plt.plot(x, y, 'b-o')
    plt.legend(['Degree'])
    plt.xlabel('$K$', fontsize = 20)
    plt.ylabel('$P_K$', fontsize = 20)
    plt.title('$Degree\,Distribution$', fontsize = 20)
    plt.show()  
    
BA= nx.random_graphs.barabasi_albert_graph(5000,2)  #生成n=20、m=1的BA无标度网络    
plotDegreeDistributionLongTail(BA)
def plotDegreeDistribution(G):
    degs = defaultdict(int)
    for i in G.degree().values(): degs[i]+=1
    items = sorted ( degs.items () )
    x, y = np.array(items).T
    x, y = np.array(items).T
    y_sum = np.sum(y)
    plt.plot(x, y, 'b-o')
    plt.xscale('log')
    plt.yscale('log')
    plt.legend(['Degree'])
    plt.xlabel('$K$', fontsize = 20)
    plt.ylabel('$P_K$', fontsize = 20)
    plt.title('$Degree\,Distribution$', fontsize = 20)
    plt.show()   
BA= nx.random_graphs.barabasi_albert_graph(50000,2)  #生成n=20、m=1的BA无标度网络        
plotDegreeDistribution(BA)
Explanation: BA网络
End of explanation
Ns = [i*10 for i in [1, 10, 100, 1000]]
ds = []
for N in Ns:
    print N
    BA= nx.random_graphs.barabasi_albert_graph(N,2)
    d = nx.average_shortest_path_length(BA)
    ds.append(d)
plt.plot(Ns, ds, 'r-o')
plt.xlabel('$N$', fontsize = 20)
plt.ylabel('$<d>$', fontsize = 20)
plt.xscale('log')
plt.show()
Explanation: 作业：
阅读 Barabasi (1999) Internet Diameter of the world wide web.Nature.401
绘制www网络的出度分布、入度分布
使用BA模型生成节点数为N、幂指数为$\gamma$的网络
计算平均路径长度d与节点数量的关系
<img src = './img/diameter.png' width = 10000>
End of explanation
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