NetworkX实战

2023-09-18 约 11238 字预计阅读 23 分钟

北京上海地铁站图数据挖掘

上海、北京地铁站点图数据挖掘，计算地铁站点的最短路径、节点重要度、集群系数、连通性。

导入工具包

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import networkx as nx
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.colors as mcolors
%matplotlib inline

可视化辅助函数

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def draw(G, pos, measures, measure_name):
    
    plt.figure(figsize=(20, 20))
    nodes = nx.draw_networkx_nodes(G, pos, node_size=250, cmap=plt.cm.plasma, 
                                   node_color=list(measures.values()),
                                   nodelist=measures.keys())
    nodes.set_norm(mcolors.SymLogNorm(linthresh=0.01, linscale=1, base=10))
    # labels = nx.draw_networkx_labels(G, pos)
    edges = nx.draw_networkx_edges(G, pos)

    plt.title(measure_name, fontsize=30)
    # plt.colorbar(nodes)
    plt.axis('off')
    plt.show()

字典按值排序辅助函数

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def dict_sort_by_value(dict_input):
    '''
    输入字典，输出按值排序的字典
    '''
    return sorted(dict_input.items(),key=lambda x : x[1], reverse=True)   

导入地铁站连接表

数据来源：

上海地铁线路图：http://www.shmetro.com

上海地铁时刻表：http://service.shmetro.com/hcskb/index.htm

北京地铁线路图：https://map.bjsubway.com

北京地铁时刻表：https://www.bjsubway.com/station/smcsj

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# 上海地铁站点连接表
df = pd.read_csv('shanghai_subway.csv')

# 北京地铁站点连接表
# df = pd.read_csv('beijing_subway.csv')

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df

	前一站	后一站	地铁线	时间（分钟）
0	莘庄	外环路	1	2
1	外环路	莲花路	1	2
2	莲花路	锦江乐园	1	3
3	锦江乐园	上海南站	1	3
4	上海南站	漕宝路	1	3
...	...	...	...	...
487	抚顺路	国权路	18	2
488	国权路	复旦大学	18	3
489	复旦大学	上海财经大学	18	2
490	上海财经大学	殷高路	18	2
491	殷高路	长江南路	18	2

492 rows × 4 columns

创建图

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# 创建无向图
G = nx.Graph()

从连接表创建图

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for idx, row in df.iterrows(): # 遍历表格的每一行
    G.add_edges_from([(row['前一站'], row['后一站'])], line=row['地铁线'], time=row['时间（分钟）'])

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# for idx, row in df.iterrows(): # 遍历表格的每一行
#     G.add_edges_from([
#         (row['前一站'], row['后一站'], {'line': row['地铁线'], 'time':row['时间（分钟）']})
#     ])

检查是否导入成功

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# 节点个数
len(G)

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# 节点个数
len(G.nodes)

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# 连接个数
len(G.edges)

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# 查看连接属性特征
G.edges[('同济大学', '四平路')]

{'line': 10, 'time': 2}

可视化

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# 节点排版布局-默认弹簧布局
pos = nx.spring_layout(G, seed=123)

# 节点排版布局-每个节点单独设置坐标
# pos = {1: [0.1, 0.9], 2: [0.4, 0.8], 3: [0.8, 0.9], 4: [0.15, 0.55],
#         5: [0.5,  0.5], 6: [0.8,  0.5], 7: [0.22, 0.3], 8: [0.30, 0.27],
#         9: [0.38, 0.24], 10: [0.7,  0.3], 11: [0.75, 0.35]}

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plt.figure(figsize=(15,15))
nx.draw(G, pos=pos)

Shortest Path 最短路径

NetworkX-最短路径算法：https://networkx.org/documentation/stable/reference/algorithms/shortest_paths.html

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# 任意两节点之间是否存在路径
nx.has_path(G, source='昌吉东路', target='同济大学')

True

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# 任意两节点之间的最短路径
nx.shortest_path(G, source='昌吉东路', target='同济大学', weight='time')

['昌吉东路',
 '上海赛车场',
 '嘉定新城',
 '马陆',
 '陈翔公路',
 '南翔',
 '桃浦新村',
 '武威路',
 '祁连山路',
 '李子园',
 '上海西站',
 '真如',
 '枫桥路',
 '曹杨路',
 '镇坪路',
 '中潭路',
 '上海火车站',
 '宝山路',
 '海伦路',
 '邮电新村',
 '四平路',
 '同济大学']

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# 任意两节点之间的最短路径长度
nx.shortest_path_length(G, source='昌吉东路', target='同济大学', weight='time')

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# 全图平均最短路径
nx.average_shortest_path_length(G, weight='time')

41.06494956638255

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# # 某一站去其他站的最短路径
# nx.single_source_shortest_path(G, source='同济大学')

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# 某一站去其他站的最短路径长度
# nx.single_source_shortest_path_length(G, source='同济大学')

地铁导航系统

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# 指定起始站和终点站
A_station = '昌吉东路'
B_station = '同济大学'

# 获取最短路径
shortest_path_list = nx.shortest_path(G, source=A_station, target=B_station, weight='time')

for i in range(len(shortest_path_list)-1):
    previous_station = shortest_path_list[i]
    next_station = shortest_path_list[i+1]
    line_id = G.edges[(previous_station, next_station)]['line'] # 地铁线编号
    time = G.edges[(previous_station, next_station)]['time']    # 时间
    print('{}--->{} {}号线 {}分钟'.format(previous_station, next_station, line_id, time)) # 输出结果
    
# 最短路径长度
print('共计 {} 分钟'.format(nx.shortest_path_length(G, source=A_station, target=B_station, weight='time')))

昌吉东路--->上海赛车场 11号线 4分钟
上海赛车场--->嘉定新城 11号线 4分钟
嘉定新城--->马陆 11号线 3分钟
马陆--->陈翔公路 11号线 4分钟
陈翔公路--->南翔 11号线 3分钟
南翔--->桃浦新村 11号线 3分钟
桃浦新村--->武威路 11号线 3分钟
武威路--->祁连山路 11号线 2分钟
祁连山路--->李子园 11号线 3分钟
李子园--->上海西站 11号线 2分钟
上海西站--->真如 11号线 3分钟
真如--->枫桥路 11号线 2分钟
枫桥路--->曹杨路 11号线 2分钟
曹杨路--->镇坪路 4号线 3分钟
镇坪路--->中潭路 4号线 2分钟
中潭路--->上海火车站 4号线 3分钟
上海火车站--->宝山路 4号线 4分钟
宝山路--->海伦路 4号线 3分钟
海伦路--->邮电新村 10号线 2分钟
邮电新村--->四平路 10号线 2分钟
四平路--->同济大学 10号线 2分钟
共计 59 分钟

Node Degree

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# dict(G.degree())

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# dict_sort_by_value(dict(G.degree()))

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draw(G, pos, dict(G.degree()), 'Node Degree')

Degree Centrality

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# nx.degree_centrality(G)

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# dict_sort_by_value(nx.degree_centrality(G))

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draw(G, pos, nx.degree_centrality(G), 'Degree Centrality')

Eigenvector Centrality（可能不收敛）

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# nx.eigenvector_centrality(G)

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dict_sort_by_value(nx.eigenvector_centrality(G))

[('南京西路', 0.34518830475920503),
 ('人民广场', 0.32714367869804883),
 ('陕西南路', 0.29476635705907395),
 ('一大会址·黄陂南路', 0.27634308858888684),
 ('静安寺', 0.27321037099703027),
 ('汉中路', 0.23555647374614153),
 ('大世界', 0.22428268234399096),
 ('常熟路', 0.18814941101791502),
 ('曲阜路', 0.17433210873324256),
 ('南京东路', 0.16614884934649515),
 ('一大会址·新天地', 0.15807115407289815),
 ('豫园', 0.1560268098793676),
 ('老西门', 0.14902071652673546),
 ('自然博物馆', 0.1433653828402798),
 ('新闸路', 0.13890944689388865),
 ('嘉善路', 0.13046505843944672),
 ('江苏路', 0.12516435598675363),
 ('淮海中路', 0.12423759493693706),
 ('肇嘉浜路', 0.12087703703752431),
 ('徐家汇', 0.10860529839818872),
 ('天潼路', 0.0972848797028609),
 ('上海图书馆', 0.0957967326770932),
 ('交通大学', 0.09326712605450173),
 ('陆家嘴', 0.09259583711035488),
 ('武定路', 0.0869184732746459),
 ('昌平路', 0.08492332314892047),
 ('隆德路', 0.08405180976485153),
 ('武宁路', 0.07885946996469545),
 ('曹杨路', 0.07765153506881935),
 ('上海火车站', 0.07677540807840737),
 ('江宁路', 0.07562534195479777),
 ('衡山路', 0.07326580131695953),
 ('马当路', 0.07228575502247306),
 ('长寿路', 0.070782329141782),
 ('陆家浜路', 0.06528067406662807),
 ('宜山路', 0.0633384339774855),
 ('大木桥路', 0.06276223702141297),
 ('东安路', 0.06238176849565273),
 ('金沙江路', 0.058751965444709536),
 ('中山公园', 0.056441377217852876),
 ('上海体育馆', 0.056272028186685856),
 ('打浦桥', 0.05005398149819857),
 ('虹桥路', 0.048190584835098596),
 ('镇坪路', 0.04729440444519273),
 ('中兴路', 0.046209625938393645),
 ('龙华中路', 0.03972543191788755),
 ('上海游泳馆', 0.03284216844978234),
 ('中潭路', 0.0306300344668945),
 ('上海体育场', 0.029297614084239275),
 ('四川北路', 0.02775006843829703),
 ('东昌路', 0.026993861657928154),
 ('漕宝路', 0.026648571941170006),
 ('浦东南路', 0.02592997856476124),
 ('国际客运中心', 0.02586095375191745),
 ('延安西路', 0.02583496933801722),
 ('西藏南路', 0.02488675862249297),
 ('宝山路', 0.024541259007936158),
 ('龙华', 0.024403908411204447),
 ('桂林路', 0.022912025965744074),
 ('枫桥路', 0.022769743149125294),
 ('中宁路', 0.022769743149125294),
 ('鲁班路', 0.021639260802360106),
 ('漕溪路', 0.020527666807312838),
 ('中山北路', 0.020274763838109527),
 ('大渡河路', 0.019805076597075946),
 ('龙漕路', 0.019788994948518823),
 ('世博会博物馆', 0.01940734737404672),
 ('娄山关路', 0.018847854611684107),
 ('小南门', 0.01824826459592638),
 ('桂林公园', 0.017013822697997696),
 ('世纪大道', 0.016756099978825333),
 ('海伦路', 0.015128915024740227),
 ('上海南站', 0.01483211996475074),
 ('真如', 0.014565008548043375),
 ('西藏北路', 0.012857640018257952),
 ('宋园路', 0.01272971743342056),
 ('岚皋路', 0.012491658380544748),
 ('浦东大道', 0.012446070401746818),
 ('后滩', 0.011346417324877846),
 ('长风公园', 0.00954424219179578),
 ('商城路', 0.00864086121103916),
 ('石龙路', 0.008550854087711803),
 ('东宝兴路', 0.007508673168681236),
 ('提篮桥', 0.007476289289692376),
 ('铜川路', 0.007249426679812461),
 ('中华艺术宫', 0.007232801738549839),
 ('梅岭北路', 0.006680680656542667),
 ('南浦大桥', 0.006657191218658478),
 ('云锦路', 0.006473532928517094),
 ('吴中路', 0.0063746443228935214),
 ('世博大道', 0.006328898598717274),
 ('长清路', 0.0062286609139391115),
 ('上海西站', 0.006195443108502219),
 ('漕河泾开发区', 0.0060533904113758644),
 ('虹口足球场', 0.005875555718867289),
 ('杨高中路', 0.005778121061354283),
 ('红宝石路', 0.005370142786911231),
 ('延长路', 0.005354109068034759),
 ('真北路', 0.00523168023322469),
 ('威宁路', 0.004978825017992365),
 ('源深体育中心', 0.004969251340305569),
 ('临平路', 0.004827039756296572),
 ('浦电路', 0.004572594456108166),
 ('虹漕路', 0.0044957409194985466),
 ('上海科技馆', 0.004480147523162711),
 ('大连路', 0.0044254553108306685),
 ('耀华路', 0.0044110460960415845),
 ('邮电新村', 0.004167930947709774),
 ('杨树浦路', 0.004164635049643304),
 ('锦江乐园', 0.003919476448894465),
 ('华东理工大学', 0.003919476448893693),
 ('源深路', 0.003569512135038515),
 ('民生路', 0.0033753044571646325),
 ('伊犁路', 0.003362845014875709),
 ('新村路', 0.0032994414915865394),
 ('成山路', 0.0031411487140143774),
 ('姚虹路', 0.0029004082761595966),
 ('塘桥', 0.0020799150248948127),
 ('昌邑路', 0.0020146473642981126),
 ('真光路', 0.0019152696299063227),
 ('曲阳路', 0.001883693955659403),
 ('龙耀路', 0.001811408766047648),
 ('蓝村路', 0.0017682199268821354),
 ('四平路', 0.001755295103356084),
 ('迎春路', 0.0017141311537059482),
 ('武威东路', 0.0016368308747279113),
 ('李子园', 0.0016368308531014269),
 ('合川路', 0.0015994233244844365),
 ('芳甸路', 0.0015628733049145526),
 ('赤峰路', 0.001551547270322036),
 ('江浦公园', 0.0014599564930694298),
 ('上海马戏城', 0.0014138912315826064),
 ('世纪公园', 0.0013937561653742923),
 ('祁连山南路', 0.00138202432056844),
 ('北新泾', 0.0013160601758069933),
 ('云台路', 0.0012637460791386503),
 ('虹梅路', 0.0011880635902502042),
 ('龙阳路', 0.0011663034785088777),
 ('东明路', 0.0010941201839890734),
 ('莲花路', 0.001035849251776343),
 ('罗秀路', 0.0010358492517732573),
 ('杨思', 0.0009883693868621057),
 ('北洋泾路', 0.0009123104804158303),
 ('水城路', 0.0008888380045468889),
 ('大华三路', 0.0008715099898583807),
 ('东方体育中心', 0.0008616466562712704),
 ('高科西路', 0.0007074848196767199),
 ('丹阳路', 0.0006242578841500306),
 ('歇浦路', 0.000592890289872722),
 ('江浦路', 0.0005891525706053762),
 ('鞍山新村', 0.0005787111669631342),
 ('蓝天路', 0.0005535268088532484),
 ('平凉路', 0.0005144548922657166),
 ('上海儿童医学中心', 0.0005107272054483918),
 ('真新新村', 0.0005060398632236523),
 ('同济大学', 0.00048045161002464577),
 ('古浪路', 0.0004324787271686449),
 ('祁连山路', 0.00043247863978375346),
 ('星中路', 0.00042263199138590753),
 ('大柏树', 0.0004097349043706069),
 ('云山路', 0.00038739071546400476),
 ('宁国路', 0.0003854984366046461),
 ('华夏中路', 0.0003846246897586962),
 ('汶水路', 0.00037337199995836015),
 ('丰庄', 0.00036499253122866546),
 ('淞虹路', 0.0003509681597168936),
 ('德平路', 0.0003207976692851701),
 ('芳华路', 0.00031998459260626375),
 ('芳芯路', 0.0003167222405210595),
 ('东兰路', 0.00031398941613808924),
 ('张江高科', 0.0003079520055746811),
 ('临沂新村', 0.0003007572216867355),
 ('高青路', 0.0002929075434238003),
 ('华鹏路', 0.00028970916901226553),
 ('花木路', 0.00028788156085499144),
 ('外环路', 0.0002737887469452818),
 ('朱梅路', 0.00027378874693370105),
 ('龙溪路', 0.00023655271725270757),
 ('灵岩南路', 0.00023260463819611013),
 ('行知路', 0.00023020664721295182),
 ('三林', 0.00022906695414238562),
 ('凌兆新村', 0.00022765380714946624),
 ('杨高南路', 0.00020678820658205882),
 ('抚顺路', 0.00019258449081970013),
 ('国权路', 0.0001910707382340992),
 ('莲溪路', 0.00015660284787605007),
 ('黄兴路', 0.00015555966731948926),
 ('台儿庄路', 0.0001461514479808473),
 ('黄杨路', 0.0001461472429082843),
 ('罗山路', 0.0001337685013604797),
 ('定边路', 0.00013371252887067506),
 ('锦绣路', 0.00013004150633398124),
 ('北中路', 0.00011683649781411802),
 ('祁安路', 0.00011427755864585485),
 ('武威路', 0.00011427722707529744),
 ('七宝', 0.00011168655559676712),
 ('江湾镇', 0.0001082992466994145),
 ('虹桥2号航站楼', 0.00010523981878059764),
 ('金桥路', 0.00010229599405730827),
 ('隆昌路', 0.00010179753191052721),
 ('中科路', 0.00010152913633164698),
 ('彭浦新村', 9.859705892814365e-05),
 ('金沙江西路', 9.599313885706678e-05),
 ('华夏西路', 9.213435265242265e-05),
 ('御桥', 8.663050138456528e-05),
 ('顾戴路', 8.296880382168908e-05),
 ('金科路', 8.131188947111461e-05),
 ('上南路', 8.020078988775062e-05),
 ('下南路', 7.918741731422785e-05),
 ('莘庄', 7.237575735460137e-05),
 ('华泾西', 7.237575731124801e-05),
 ('上海动物园', 6.905084145021053e-05),
 ('三林东', 6.587483382235718e-05),
 ('大场镇', 6.0810735633317576e-05),
 ('芦恒路', 6.015221993516688e-05),
 ('复旦大学', 5.059116545983654e-05),
 ('五角场', 5.045024325873357e-05),
 ('陈春路', 4.631030751906198e-05),
 ('虹桥1号航站楼', 4.30441410805867e-05),
 ('延吉中路', 4.107282131124138e-05),
 ('金桥', 3.8602629952531715e-05),
 ('云顺路', 3.8585594926562154e-05),
 ('浦三路', 3.7658616676167844e-05),
 ('嘉怡路', 3.5333693341064326e-05),
 ('周浦东', 3.5325790612266245e-05),
 ('秀沿路', 3.531460528829171e-05),
 ('虹桥火车站', 3.214643341063452e-05),
 ('北蔡', 3.0987072791060275e-05),
 ('南大路', 3.0200381450407823e-05),
 ('桃浦新村', 3.019912827307753e-05),
 ('中春路', 2.9517885308330953e-05),
 ('殷高西路', 2.8993571989767872e-05),
 ('博兴路', 2.706829176293026e-05),
 ('爱国路', 2.6917645251336415e-05),
 ('学林路', 2.668845362396324e-05),
 ('共康路', 2.603646829290025e-05),
 ('金运路', 2.3707557375910697e-05),
 ('康桥', 2.2880578017653626e-05),
 ('虹莘路', 2.183396879575416e-05),
 ('广兰路', 2.1469814539144915e-05),
 ('春申路', 1.913543471800008e-05),
 ('虹梅南路', 1.9135434555495407e-05),
 ('场中路', 1.6064940238762195e-05),
 ('浦江镇', 1.5894843290796643e-05),
 ('上海财经大学', 1.389145200192103e-05),
 ('江湾体育场', 1.3320507989667622e-05),
 ('黄兴公园', 1.0844032390498784e-05),
 ('金吉路', 1.024895634040226e-05),
 ('浦东足球场', 1.0184146988913355e-05),
 ('临洮路', 9.3351708704171e-06),
 ('鹤沙航城', 9.329671887799534e-06),
 ('康新公路', 9.284415930638734e-06),
 ('长江南路', 9.157809732431543e-06),
 ('诸光路', 8.493984750404255e-06),
 ('龙柏新村', 8.491290081546015e-06),
 ('丰翔路', 7.986065347070663e-06),
 ('南翔', 7.981328901681124e-06),
 ('徐泾东', 7.939542480126271e-06),
 ('九亭', 7.802299209891346e-06),
 ('五莲路', 7.374855805492639e-06),
 ('复兴岛', 7.257261233416775e-06),
 ('通河新村', 6.875359111788295e-06),
 ('张江路', 6.58817740797362e-06),
 ('周浦', 6.0439405959160685e-06),
 ('殷高路', 5.68913878863468e-06),
 ('唐镇', 5.669062135147163e-06),
 ('七莘路', 5.395593476698256e-06),
 ('银都路', 5.060132186224369e-06),
 ('景西路', 5.060131576269231e-06),
 ('上大路', 4.246791633419134e-06),
 ('江月路', 4.199284519605183e-06),
 ('三门路', 3.516942151319783e-06),
 ('金海路', 2.922160108332692e-06),
 ('翔殷路', 2.862108404036366e-06),
 ('巨峰路', 2.812373402257501e-06),
 ('金粤路', 2.6766187609425924e-06),
 ('东陆路', 2.4853186757134015e-06),
 ('航头东', 2.4643100499360915e-06),
 ('乐秀路', 2.456211272228161e-06),
 ('淞发路', 2.418017752550578e-06),
 ('迪士尼', 2.2923268717626103e-06),
 ('蟠龙路', 2.2445378510100166e-06),
 ('紫藤路', 2.233634307270689e-06),
 ('锦秋路', 2.1275517941027036e-06),
 ('陈翔公路', 2.1096458102969884e-06),
 ('泗泾', 2.0626298860041646e-06),
 ('呼兰路', 1.815526345079715e-06),
 ('繁荣路', 1.5968157450142117e-06),
 ('创新中路', 1.4969618823401352e-06),
 ('颛桥', 1.338369412987595e-06),
 ('曙建路', 1.338367120825213e-06),
 ('南陈路', 1.1323857696790865e-06),
 ('联航路', 1.1048489132148824e-06),
 ('殷高东路', 9.285278822875906e-07),
 ('申江路', 8.193989270942173e-07),
 ('杨高北路', 7.922930588010286e-07),
 ('顾唐路', 7.712500607519193e-07),
 ('嫩江路', 7.522042423411743e-07),
 ('东靖路', 7.424977244597947e-07),
 ('桂桥路', 6.606161442250491e-07),
 ('新场', 6.51034021652862e-07),
 ('张华浜', 6.384325713646042e-07),
 ('顾村公园', 6.254152790687604e-07),
 ('封浜', 6.06874735333989e-07),
 ('徐盈路', 5.931779898987262e-07),
 ('马陆', 5.577069963479668e-07),
 ('航中路', 5.517476014197456e-07),
 ('佘山', 5.453696697009482e-07),
 ('共富新村', 4.793960231362631e-07),
 ('沈梅路', 4.219843712816389e-07),
 ('金京路', 3.977829250264296e-07),
 ('华夏东路', 3.9531027766563646e-07),
 ('北桥', 3.540747574772615e-07),
 ('双柏路', 3.5406613486469205e-07),
 ('上海大学', 3.385229218199973e-07),
 ('沈杜公路', 2.7297364509663744e-07),
 ('新江湾城', 2.4513691583190427e-07),
 ('祁华路', 2.381744420948116e-07),
 ('民雷路', 2.0270142913861996e-07),
 ('五洲大道', 1.9603039710651717e-07),
 ('市光路', 1.856434794471213e-07),
 ('野生动物园', 1.7203838559130252e-07),
 ('淞滨路', 1.685607318622697e-07),
 ('刘行', 1.653625126136348e-07),
 ('徐泾北城', 1.5678068428999187e-07),
 ('嘉定新城', 1.474602382775197e-07),
 ('洞泾', 1.442257188993948e-07),
 ('宝安公路', 1.2654689042081076e-07),
 ('鹤涛路', 1.1152432880799095e-07),
 ('川沙', 1.0440193856554608e-07),
 ('剑川路', 9.369904774262517e-08),
 ('元江路', 9.366658252213859e-08),
 ('国帆路', 6.47161144465196e-08),
 ('洲海路', 5.175810533707712e-08),
 ('曹路', 5.002963760098825e-08),
 ('惠南', 4.547848857949125e-08),
 ('水产路', 4.4502148043878555e-08),
 ('潘广路', 4.3718166159875194e-08),
 ('嘉松中路', 4.144359229612825e-08),
 ('白银路', 3.899669573711562e-08),
 ('松江大学城', 3.8149491837363796e-08),
 ('友谊西路', 3.326351669368966e-08),
 ('下沙', 2.9363434860805047e-08),
 ('凌空路', 2.7576280942869793e-08),
 ('东川路', 2.4803650989623975e-08),
 ('永德路', 2.46813212649182e-08),
 ('双江路', 1.708895506049985e-08),
 ('外高桥保税区南站', 1.3672268341756097e-08),
 ('惠南东', 1.2028167424890085e-08),
 ('宝杨路', 1.1748385691452752e-08),
 ('罗南新村', 1.1511626793160564e-08),
 ('赵巷', 1.0956924381367109e-08),
 ('嘉定西', 1.0315218108832666e-08),
 ('松江新城', 1.0093352857726008e-08),
 ('富锦路', 8.210622560671108e-09),
 ('远东大道', 7.283323035094685e-09),
 ('航头', 7.261958216822991e-09),
 ('江川路', 6.5683787771182e-09),
 ('紫竹高新区', 6.1071303290981765e-09),
 ('高桥西', 4.5300828131546704e-09),
 ('航津路', 3.6335154149788168e-09),
 ('书院', 3.1824816648712607e-09),
 ('友谊路', 3.100468256406229e-09),
 ('汇金路', 2.8973229616244033e-09),
 ('美兰湖', 2.8464308457705746e-09),
 ('嘉定北', 2.7292461471009787e-09),
 ('松江体育中心', 2.670415798052803e-09),
 ('海天三路', 1.916073159712836e-09),
 ('西渡', 1.740156144517946e-09),
 ('高桥', 1.2681133827697382e-09),
 ('外高桥保税区北站', 1.0472670808864178e-09),
 ('临港大道', 8.392009147533852e-10),
 ('铁力路', 8.147483303301796e-10),
 ('青浦新城', 7.662911149570269e-10),
 ('上海赛车场', 7.223360738663101e-10),
 ('醉白池', 7.037382414795119e-10),
 ('港城路', 6.086085102874774e-10),
 ('浦东国际机场', 4.734473543870188e-10),
 ('萧塘', 4.612484483959065e-10),
 ('滴水湖', 2.0790492492163032e-10),
 ('漕盈路', 2.0271460316168133e-10),
 ('江杨北路', 2.0105418042463568e-10),
 ('昌吉东路', 1.912449298550137e-10),
 ('松江南站', 1.7415177584811688e-10),
 ('基隆路', 1.5024096205463907e-10),
 ('奉浦大道', 1.2232972542206332e-10),
 ('淀山湖大道', 5.3624867368877837e-11),
 ('上海汽车城', 5.065466344149559e-11),
 ('环城东路', 3.246522146891751e-11),
 ('朱家角', 1.4129609700584414e-11),
 ('安亭', 1.3422980238458734e-11),
 ('望园路', 8.622613686296197e-12),
 ('兆丰路', 3.5578715548799247e-12),
 ('东方绿舟', 3.49607594627485e-12),
 ('金海湖', 2.2921633664651897e-12),
 ('光明路', 9.395865630498605e-13),
 ('奉贤新城', 6.09955284192513e-13),
 ('花桥', 2.330088423693451e-13),
 ('金平路', 1.6250169546464013e-13),
 ('华宁路', 4.333913458088084e-14),
 ('文井路', 1.1525417889624957e-14),
 ('闵行开发区', 2.8780955283508334e-15)]

1

draw(G, pos, nx.eigenvector_centrality(G), 'Eigenvector Centrality')

Betweenness Centrality（必经之地）

1

# nx.betweenness_centrality(G)

1

# dict_sort_by_value(nx.betweenness_centrality(G))

1

draw(G, pos, nx.betweenness_centrality(G), 'Betweenness Centrality')

Closeness Centrality（去哪儿都近）

1

# nx.closeness_centrality(G)

1

# dict_sort_by_value(nx.eigenvector_centrality(G))

1

draw(G, pos, nx.closeness_centrality(G), 'Closeness Centrality')

PageRank

1

# nx.pagerank(G)

1

# dict_sort_by_value(nx.pagerank(G))

1

draw(G, pos, nx.pagerank(G, alpha=0.85), 'PageRank')

Katz Centrality

1

# nx.katz_centrality(G, alpha=0.1, beta=1.0)

1

# dict_sort_by_value(nx.katz_centrality(G, alpha=0.1, beta=1.0))

1

draw(G, pos, nx.katz_centrality(G, alpha=0.1, beta=1.0), 'Katz Centrality')

HITS Hubs and Authorities

1
2
3


h, a = nx.hits(G)
draw(G, pos, h, 'DiGraph HITS Hubs')
draw(G, pos, a, 'DiGraph HITS Authorities')

NetworkX文档：社群属性 Clustering

https://networkx.org/documentation/stable/reference/algorithms/clustering.html

三角形个数

1

# nx.triangles(G)

1

dict_sort_by_value(nx.triangles(G))

[('曹杨路', 2),
 ('大世界', 2),
 ('豫园', 2),
 ('隆德路', 2),
 ('上海南站', 1),
 ('漕宝路', 1),
 ('上海体育馆', 1),
 ('徐家汇', 1),
 ('一大会址·黄陂南路', 1),
 ('人民广场', 1),
 ('汉中路', 1),
 ('南京西路', 1),
 ('南京东路', 1),
 ('陆家嘴', 1),
 ('宜山路', 1),
 ('金沙江路', 1),
 ('大木桥路', 1),
 ('东安路', 1),
 ('龙华中路', 1),
 ('长清路', 1),
 ('耀华路', 1),
 ('成山路', 1),
 ('老西门', 1),
 ('上海西站', 1),
 ('真如', 1),
 ('桂林公园', 1),
 ('武宁路', 1),
 ('自然博物馆', 1),
 ('铜川路', 1),
 ('莘庄', 0),
 ('外环路', 0),
 ('莲花路', 0),
 ('锦江乐园', 0),
 ('衡山路', 0),
 ('常熟路', 0),
 ('陕西南路', 0),
 ('新闸路', 0),
 ('上海火车站', 0),
 ('中山北路', 0),
 ('延长路', 0),
 ('上海马戏城', 0),
 ('汶水路', 0),
 ('彭浦新村', 0),
 ('共康路', 0),
 ('通河新村', 0),
 ('呼兰路', 0),
 ('共富新村', 0),
 ('宝安公路', 0),
 ('友谊西路', 0),
 ('富锦路', 0),
 ('徐泾东', 0),
 ('虹桥火车站', 0),
 ('虹桥2号航站楼', 0),
 ('淞虹路', 0),
 ('北新泾', 0),
 ('威宁路', 0),
 ('娄山关路', 0),
 ('中山公园', 0),
 ('江苏路', 0),
 ('静安寺', 0),
 ('东昌路', 0),
 ('世纪大道', 0),
 ('上海科技馆', 0),
 ('世纪公园', 0),
 ('龙阳路', 0),
 ('张江高科', 0),
 ('金科路', 0),
 ('广兰路', 0),
 ('唐镇', 0),
 ('创新中路', 0),
 ('华夏东路', 0),
 ('川沙', 0),
 ('凌空路', 0),
 ('远东大道', 0),
 ('海天三路', 0),
 ('浦东国际机场', 0),
 ('石龙路', 0),
 ('龙漕路', 0),
 ('漕溪路', 0),
 ('虹桥路', 0),
 ('延安西路', 0),
 ('镇坪路', 0),
 ('中潭路', 0),
 ('宝山路', 0),
 ('东宝兴路', 0),
 ('虹口足球场', 0),
 ('赤峰路', 0),
 ('大柏树', 0),
 ('江湾镇', 0),
 ('殷高西路', 0),
 ('长江南路', 0),
 ('淞发路', 0),
 ('张华浜', 0),
 ('淞滨路', 0),
 ('水产路', 0),
 ('宝杨路', 0),
 ('友谊路', 0),
 ('铁力路', 0),
 ('江杨北路', 0),
 ('海伦路', 0),
 ('临平路', 0),
 ('大连路', 0),
 ('杨树浦路', 0),
 ('浦东大道', 0),
 ('浦电路', 0),
 ('蓝村路', 0),
 ('塘桥', 0),
 ('南浦大桥', 0),
 ('西藏南路', 0),
 ('鲁班路', 0),
 ('上海体育场', 0),
 ('春申路', 0),
 ('银都路', 0),
 ('颛桥', 0),
 ('北桥', 0),
 ('剑川路', 0),
 ('东川路', 0),
 ('江川路', 0),
 ('西渡', 0),
 ('萧塘', 0),
 ('奉浦大道', 0),
 ('环城东路', 0),
 ('望园路', 0),
 ('金海湖', 0),
 ('奉贤新城', 0),
 ('金平路', 0),
 ('华宁路', 0),
 ('文井路', 0),
 ('闵行开发区', 0),
 ('东方体育中心', 0),
 ('灵岩南路', 0),
 ('上南路', 0),
 ('华夏西路', 0),
 ('高青路', 0),
 ('东明路', 0),
 ('高科西路', 0),
 ('临沂新村', 0),
 ('上海儿童医学中心', 0),
 ('源深体育中心', 0),
 ('民生路', 0),
 ('北洋泾路', 0),
 ('德平路', 0),
 ('云山路', 0),
 ('金桥路', 0),
 ('博兴路', 0),
 ('五莲路', 0),
 ('巨峰路', 0),
 ('东靖路', 0),
 ('五洲大道', 0),
 ('洲海路', 0),
 ('外高桥保税区南站', 0),
 ('航津路', 0),
 ('外高桥保税区北站', 0),
 ('港城路', 0),
 ('美兰湖', 0),
 ('罗南新村', 0),
 ('潘广路', 0),
 ('刘行', 0),
 ('顾村公园', 0),
 ('祁华路', 0),
 ('上海大学', 0),
 ('南陈路', 0),
 ('上大路', 0),
 ('场中路', 0),
 ('大场镇', 0),
 ('行知路', 0),
 ('大华三路', 0),
 ('新村路', 0),
 ('岚皋路', 0),
 ('长寿路', 0),
 ('昌平路', 0),
 ('肇嘉浜路', 0),
 ('后滩', 0),
 ('云台路', 0),
 ('杨高南路', 0),
 ('锦绣路', 0),
 ('芳华路', 0),
 ('花木路', 0),
 ('沈杜公路', 0),
 ('联航路', 0),
 ('江月路', 0),
 ('浦江镇', 0),
 ('芦恒路', 0),
 ('凌兆新村', 0),
 ('杨思', 0),
 ('中华艺术宫', 0),
 ('陆家浜路', 0),
 ('曲阜路', 0),
 ('中兴路', 0),
 ('西藏北路', 0),
 ('曲阳路', 0),
 ('四平路', 0),
 ('鞍山新村', 0),
 ('江浦路', 0),
 ('黄兴路', 0),
 ('延吉中路', 0),
 ('黄兴公园', 0),
 ('翔殷路', 0),
 ('嫩江路', 0),
 ('市光路', 0),
 ('松江南站', 0),
 ('醉白池', 0),
 ('松江体育中心', 0),
 ('松江新城', 0),
 ('松江大学城', 0),
 ('洞泾', 0),
 ('佘山', 0),
 ('泗泾', 0),
 ('九亭', 0),
 ('中春路', 0),
 ('七宝', 0),
 ('星中路', 0),
 ('合川路', 0),
 ('漕河泾开发区', 0),
 ('桂林路', 0),
 ('嘉善路', 0),
 ('打浦桥', 0),
 ('马当路', 0),
 ('小南门', 0),
 ('商城路', 0),
 ('杨高中路', 0),
 ('芳甸路', 0),
 ('蓝天路', 0),
 ('台儿庄路', 0),
 ('金桥', 0),
 ('金吉路', 0),
 ('金海路', 0),
 ('顾唐路', 0),
 ('民雷路', 0),
 ('曹路', 0),
 ('航中路', 0),
 ('紫藤路', 0),
 ('龙柏新村', 0),
 ('虹桥1号航站楼', 0),
 ('上海动物园', 0),
 ('龙溪路', 0),
 ('水城路', 0),
 ('伊犁路', 0),
 ('宋园路', 0),
 ('交通大学', 0),
 ('上海图书馆', 0),
 ('一大会址·新天地', 0),
 ('天潼路', 0),
 ('四川北路', 0),
 ('邮电新村', 0),
 ('同济大学', 0),
 ('国权路', 0),
 ('五角场', 0),
 ('江湾体育场', 0),
 ('三门路', 0),
 ('殷高东路', 0),
 ('新江湾城', 0),
 ('国帆路', 0),
 ('双江路', 0),
 ('高桥西', 0),
 ('高桥', 0),
 ('基隆路', 0),
 ('花桥', 0),
 ('光明路', 0),
 ('兆丰路', 0),
 ('安亭', 0),
 ('上海汽车城', 0),
 ('昌吉东路', 0),
 ('上海赛车场', 0),
 ('嘉定北', 0),
 ('嘉定西', 0),
 ('白银路', 0),
 ('嘉定新城', 0),
 ('马陆', 0),
 ('陈翔公路', 0),
 ('南翔', 0),
 ('桃浦新村', 0),
 ('武威路', 0),
 ('祁连山路', 0),
 ('李子园', 0),
 ('枫桥路', 0),
 ('上海游泳馆', 0),
 ('龙华', 0),
 ('云锦路', 0),
 ('龙耀路', 0),
 ('三林', 0),
 ('三林东', 0),
 ('浦三路', 0),
 ('御桥', 0),
 ('罗山路', 0),
 ('秀沿路', 0),
 ('康新公路', 0),
 ('迪士尼', 0),
 ('七莘路', 0),
 ('虹莘路', 0),
 ('顾戴路', 0),
 ('东兰路', 0),
 ('虹梅路', 0),
 ('虹漕路', 0),
 ('国际客运中心', 0),
 ('提篮桥', 0),
 ('江浦公园', 0),
 ('宁国路', 0),
 ('隆昌路', 0),
 ('爱国路', 0),
 ('复兴岛', 0),
 ('东陆路', 0),
 ('杨高北路', 0),
 ('金京路', 0),
 ('申江路', 0),
 ('金运路', 0),
 ('金沙江西路', 0),
 ('丰庄', 0),
 ('祁连山南路', 0),
 ('真北路', 0),
 ('大渡河路', 0),
 ('江宁路', 0),
 ('淮海中路', 0),
 ('世博会博物馆', 0),
 ('世博大道', 0),
 ('华鹏路', 0),
 ('下南路', 0),
 ('北蔡', 0),
 ('陈春路', 0),
 ('莲溪路', 0),
 ('华夏中路', 0),
 ('中科路', 0),
 ('学林路', 0),
 ('张江路', 0),
 ('封浜', 0),
 ('乐秀路', 0),
 ('临洮路', 0),
 ('嘉怡路', 0),
 ('定边路', 0),
 ('真新新村', 0),
 ('真光路', 0),
 ('中宁路', 0),
 ('武定路', 0),
 ('浦东南路', 0),
 ('源深路', 0),
 ('昌邑路', 0),
 ('歇浦路', 0),
 ('黄杨路', 0),
 ('云顺路', 0),
 ('浦东足球场', 0),
 ('金粤路', 0),
 ('桂桥路', 0),
 ('紫竹高新区', 0),
 ('永德路', 0),
 ('元江路', 0),
 ('双柏路', 0),
 ('曙建路', 0),
 ('景西路', 0),
 ('虹梅南路', 0),
 ('华泾西', 0),
 ('朱梅路', 0),
 ('罗秀路', 0),
 ('华东理工大学', 0),
 ('吴中路', 0),
 ('姚虹路', 0),
 ('红宝石路', 0),
 ('长风公园', 0),
 ('梅岭北路', 0),
 ('武威东路', 0),
 ('古浪路', 0),
 ('祁安路', 0),
 ('南大路', 0),
 ('丰翔路', 0),
 ('锦秋路', 0),
 ('周浦东', 0),
 ('鹤沙航城', 0),
 ('航头东', 0),
 ('新场', 0),
 ('野生动物园', 0),
 ('惠南', 0),
 ('惠南东', 0),
 ('书院', 0),
 ('临港大道', 0),
 ('滴水湖', 0),
 ('诸光路', 0),
 ('蟠龙路', 0),
 ('徐盈路', 0),
 ('徐泾北城', 0),
 ('嘉松中路', 0),
 ('赵巷', 0),
 ('汇金路', 0),
 ('青浦新城', 0),
 ('漕盈路', 0),
 ('淀山湖大道', 0),
 ('朱家角', 0),
 ('东方绿舟', 0),
 ('航头', 0),
 ('下沙', 0),
 ('鹤涛路', 0),
 ('沈梅路', 0),
 ('繁荣路', 0),
 ('周浦', 0),
 ('康桥', 0),
 ('北中路', 0),
 ('芳芯路', 0),
 ('迎春路', 0),
 ('丹阳路', 0),
 ('平凉路', 0),
 ('抚顺路', 0),
 ('复旦大学', 0),
 ('上海财经大学', 0),
 ('殷高路', 0)]

1

draw(G, pos, nx.triangles(G), 'Triangles')

Clustering Coefficient

1

# nx.clustering(G)

1

# dict_sort_by_value(nx.clustering(G))

1

draw(G, pos, nx.clustering(G), 'Clustering Coefficient')

计算全图Graphlet个数

导入工具包

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5


import networkx as nx
import matplotlib.pyplot as plt
%matplotlib inline

import itertools

导入全图

1

G = nx.karate_club_graph()

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2
3


plt.figure(figsize=(10,8))
pos = nx.spring_layout(G, seed=123)
nx.draw(G, pos, with_labels=True)

指定Graphlet

1

target = nx.complete_graph(3)

1

nx.draw(target)

匹配Graphlet，统计个数

1
2
3
4
5
6


num = 0
for sub_nodes in itertools.combinations(G.nodes(), len(target.nodes())):  # 遍历全图中，符合graphlet节点个数的所有节点组合
    subg = G.subgraph(sub_nodes)                                          # 从全图中抽取出子图
    if nx.is_connected(subg) and nx.is_isomorphic(subg, target):          # 如果子图是完整连通域，并且符合graphlet特征，输出原图节点编号
        num += 1
        print(subg.edges())

[(0, 1), (0, 2), (1, 2)]
[(0, 1), (0, 3), (1, 3)]
[(0, 1), (0, 7), (1, 7)]
[(0, 1), (0, 13), (1, 13)]
[(0, 1), (0, 17), (1, 17)]
[(0, 1), (0, 19), (1, 19)]
[(0, 1), (0, 21), (1, 21)]
[(0, 2), (0, 3), (2, 3)]
[(0, 2), (0, 7), (2, 7)]
[(0, 2), (0, 8), (8, 2)]
[(0, 2), (0, 13), (2, 13)]
[(0, 3), (0, 7), (3, 7)]
[(0, 3), (0, 12), (3, 12)]
[(0, 3), (0, 13), (3, 13)]
[(0, 4), (0, 6), (4, 6)]
[(0, 4), (0, 10), (10, 4)]
[(0, 5), (0, 6), (5, 6)]
[(0, 5), (0, 10), (10, 5)]
[(1, 2), (1, 3), (2, 3)]
[(1, 2), (1, 7), (2, 7)]
[(1, 2), (1, 13), (2, 13)]
[(1, 3), (1, 7), (3, 7)]
[(1, 3), (1, 13), (3, 13)]
[(2, 3), (2, 7), (3, 7)]
[(2, 3), (2, 13), (3, 13)]
[(8, 2), (8, 32), (2, 32)]
[(16, 5), (16, 6), (5, 6)]
[(8, 30), (8, 32), (32, 30)]
[(8, 30), (8, 33), (33, 30)]
[(8, 32), (8, 33), (33, 32)]
[(32, 14), (32, 33), (33, 14)]
[(32, 15), (32, 33), (33, 15)]
[(32, 18), (32, 33), (33, 18)]
[(32, 20), (32, 33), (33, 20)]
[(32, 22), (32, 33), (33, 22)]
[(33, 23), (33, 27), (27, 23)]
[(32, 23), (32, 29), (29, 23)]
[(33, 23), (33, 29), (29, 23)]
[(32, 23), (32, 33), (33, 23)]
[(24, 25), (24, 31), (25, 31)]
[(33, 26), (33, 29), (26, 29)]
[(33, 28), (33, 31), (28, 31)]
[(32, 29), (32, 33), (33, 29)]
[(32, 30), (32, 33), (33, 30)]
[(32, 31), (32, 33), (33, 31)]

1

num

拉普拉斯矩阵特征值分解

参考文档：https://networkx.org/documentation/stable/auto_examples/drawing/plot_eigenvalues.html#sphx-glr-auto-examples-drawing-plot-eigenvalues-py

导入工具包

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10
11


# 图数据挖掘
import networkx as nx

import numpy as np

# 数据可视化
import matplotlib.pyplot as plt
%matplotlib inline

plt.rcParams['font.sans-serif']=['SimHei']  # 用来正常显示中文标签  
plt.rcParams['axes.unicode_minus']=False  # 用来正常显示负号

1

import numpy.linalg # 线性代数

创建图

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n = 1000  # 节点个数
m = 5000  # 连接个数
G = nx.gnm_random_graph(n, m, seed=5040)

邻接矩阵（Adjacency Matrix）

1

A = nx.adjacency_matrix(G)

1

A.shape

(1000, 1000)

1

A.todense()

matrix([[0, 0, 0, ..., 0, 0, 0],
        [0, 0, 0, ..., 0, 0, 0],
        [0, 0, 0, ..., 0, 0, 0],
        ...,
        [0, 0, 0, ..., 0, 0, 0],
        [0, 0, 0, ..., 0, 0, 0],
        [0, 0, 0, ..., 0, 0, 0]], dtype=int64)

拉普拉斯矩阵（Laplacian Matrix）

$$ L = D - A $$

L 为拉普拉斯矩阵（Laplacian Matrix）

D 为节点degree对角矩阵

A 为邻接矩阵（Adjacency Matrix）

1

L = nx.laplacian_matrix(G)

1

L.shape

(1000, 1000)

1
2


# 节点degree对角矩阵
D = L + A

1

D.todense()

matrix([[12,  0,  0, ...,  0,  0,  0],
        [ 0,  6,  0, ...,  0,  0,  0],
        [ 0,  0,  8, ...,  0,  0,  0],
        ...,
        [ 0,  0,  0, ...,  8,  0,  0],
        [ 0,  0,  0, ...,  0,  6,  0],
        [ 0,  0,  0, ...,  0,  0,  7]], dtype=int64)

1

# G.degree()

归一化拉普拉斯矩阵（Normalized Laplacian Matrix）

$$ L_n = D^{-\frac{1}{2}}LD^{-\frac{1}{2}} $$

1

L_n = nx.normalized_laplacian_matrix(G)

1

L_n.shape

(1000, 1000)

1

L_n.todense()

matrix([[1., 0., 0., ..., 0., 0., 0.],
        [0., 1., 0., ..., 0., 0., 0.],
        [0., 0., 1., ..., 0., 0., 0.],
        ...,
        [0., 0., 0., ..., 1., 0., 0.],
        [0., 0., 0., ..., 0., 1., 0.],
        [0., 0., 0., ..., 0., 0., 1.]])

1
2


plt.imshow(L_n.todense())
plt.show()

1

type(L_n)

scipy.sparse.csr.csr_matrix

特征值分解

1

e = np.linalg.eigvals(L_n.A)

1

e

array([-3.53879524e-16,  1.59246179e+00,  1.58938840e+00,  1.58583526e+00,
        4.02846420e-01,  4.06224295e-01,  4.12963520e-01,  4.15548851e-01,
        4.16517467e-01,  1.58139953e+00,  1.58070318e+00,  4.20582471e-01,
        4.24326947e-01,  1.57786096e+00,  4.29110624e-01,  4.31464217e-01,
        1.57398179e+00,  1.57237733e+00,  4.33218297e-01,  1.57150969e+00,
        1.57095514e+00,  4.35026221e-01,  4.37026657e-01,  1.56726875e+00,
        4.40378510e-01,  4.42775893e-01,  4.45046086e-01,  1.56345866e+00,
        1.56395825e+00,  4.46753827e-01,  4.50881547e-01,  4.48717950e-01,
        4.49417823e-01,  1.56009824e+00,  1.55880096e+00,  1.55717189e+00,
        1.55698938e+00,  1.55475549e+00,  1.54908738e+00,  1.54681927e+00,
        1.54491505e+00,  1.54352783e+00,  1.54231896e+00,  1.54098279e+00,
        1.54051439e+00,  1.53578067e+00,  1.53502733e+00,  1.53439005e+00,
        1.52838134e+00,  1.53085674e+00,  1.53141471e+00,  1.52515680e+00,
        1.52512114e+00,  1.52151217e+00,  1.52011306e+00,  1.52064972e+00,
        1.51734481e+00,  1.51789644e+00,  1.51650845e+00,  1.51050284e+00,
        1.51160361e+00,  1.51266269e+00,  1.50970637e+00,  1.50786560e+00,
        1.50593131e+00,  1.50508573e+00,  1.50356363e+00,  1.50256425e+00,
        1.50143743e+00,  1.49974932e+00,  1.50033534e+00,  1.48749487e+00,
        1.48806489e+00,  1.49053530e+00,  1.49336377e+00,  1.49505706e+00,
        1.49650057e+00,  1.49229941e+00,  1.49722359e+00,  1.48940564e+00,
        4.54190661e-01,  4.55357875e-01,  4.58263114e-01,  4.57707915e-01,
        4.61849308e-01,  4.63660933e-01,  4.64660065e-01,  4.66860929e-01,
        4.69366311e-01,  4.69709000e-01,  4.73545468e-01,  4.76756569e-01,
        4.65763726e-01,  4.74333012e-01,  4.86462532e-01,  4.79394325e-01,
        4.75649819e-01,  4.80606513e-01,  4.83264820e-01,  4.84997389e-01,
        4.85154008e-01,  1.48542791e+00,  1.48490224e+00,  1.47887385e+00,
        1.48185134e+00,  1.48241248e+00,  1.47794501e+00,  1.47709645e+00,
        1.47762024e+00,  1.47552207e+00,  1.47141067e+00,  1.47388061e+00,
        1.47033271e+00,  1.44904512e+00,  1.44986013e+00,  1.45090139e+00,
        1.45216238e+00,  1.46817150e+00,  1.46566928e+00,  1.46791556e+00,
        1.45500080e+00,  1.46487844e+00,  1.46269200e+00,  1.45908676e+00,
        1.46138525e+00,  1.45410520e+00,  1.45733801e+00,  1.46156481e+00,
        1.45644004e+00,  1.45770660e+00,  4.91621993e-01,  4.90304444e-01,
        4.88729605e-01,  4.88322850e-01,  4.94845030e-01,  4.96583277e-01,
        5.00214562e-01,  5.02653227e-01,  4.95732976e-01,  5.04153801e-01,
        4.99448904e-01,  5.06512201e-01,  5.05556356e-01,  5.01180333e-01,
        1.42699849e+00,  1.42677234e+00,  1.42988927e+00,  1.43134759e+00,
        1.43260357e+00,  1.43846236e+00,  1.44014343e+00,  1.44739972e+00,
        1.43619292e+00,  1.44246039e+00,  1.44531749e+00,  1.43354375e+00,
        1.43441916e+00,  1.44589494e+00,  1.43490982e+00,  1.43713890e+00,
        1.44158907e+00,  1.44376226e+00,  5.22556927e-01,  5.09010234e-01,
        5.10154623e-01,  5.11622478e-01,  5.07002615e-01,  5.16057435e-01,
        5.14665415e-01,  5.16533531e-01,  5.18785357e-01,  5.20015126e-01,
        5.21860072e-01,  5.12797358e-01,  1.41228135e+00,  1.41347984e+00,
        1.41448341e+00,  1.41598484e+00,  1.41748057e+00,  1.41662929e+00,
        1.42320539e+00,  1.42542313e+00,  1.41961351e+00,  1.42457194e+00,
        1.42052766e+00,  1.42118742e+00,  5.23407726e-01,  5.34473212e-01,
        5.25881593e-01,  5.27322194e-01,  5.30802877e-01,  5.29711336e-01,
        5.32286178e-01,  5.28274379e-01,  5.33613821e-01,  1.39936317e+00,
        1.40196461e+00,  1.40361810e+00,  1.40415543e+00,  1.40498931e+00,
        1.40702610e+00,  1.41140670e+00,  1.40819911e+00,  1.40969057e+00,
        1.41274341e+00,  5.44579715e-01,  5.39616555e-01,  5.37113507e-01,
        5.38466485e-01,  5.40543078e-01,  5.42474931e-01,  5.42041958e-01,
        1.38748697e+00,  1.38793243e+00,  1.39757348e+00,  1.38899394e+00,
        1.39625743e+00,  1.39026791e+00,  1.39235250e+00,  1.39158171e+00,
        1.39285233e+00,  1.39073914e+00,  1.39651349e+00,  5.48067206e-01,
        5.55908434e-01,  5.47076780e-01,  5.45647110e-01,  5.51103949e-01,
        5.51314655e-01,  5.53584120e-01,  5.54656573e-01,  1.37512858e+00,
        1.37634495e+00,  1.37774739e+00,  1.37947093e+00,  1.38336173e+00,
        1.38561715e+00,  1.37881800e+00,  1.38125128e+00,  1.38474073e+00,
        1.38167394e+00,  5.57471020e-01,  5.58337114e-01,  5.60822980e-01,
        5.63088387e-01,  5.63882730e-01,  5.65472145e-01,  5.57092578e-01,
        5.60005758e-01,  5.59842586e-01,  1.36388306e+00,  1.36516216e+00,
        1.36665422e+00,  1.37131925e+00,  1.37392113e+00,  1.36916170e+00,
        1.36865728e+00,  1.37257584e+00,  1.36949557e+00,  5.66792887e-01,
        5.76150660e-01,  5.68163292e-01,  5.71180265e-01,  5.73122697e-01,
        5.69767689e-01,  5.71671829e-01,  5.75206362e-01,  5.74332679e-01,
        1.35156900e+00,  1.35260850e+00,  1.35396561e+00,  1.35207043e+00,
        1.35825257e+00,  1.35879561e+00,  1.35989991e+00,  1.35586993e+00,
        1.36203150e+00,  1.36246682e+00,  1.36309440e+00,  5.77960813e-01,
        5.79394345e-01,  5.87460119e-01,  5.83681701e-01,  5.79719792e-01,
        5.80876426e-01,  5.84596718e-01,  5.85274871e-01,  5.86873671e-01,
        5.80144591e-01,  1.33708760e+00,  1.33958685e+00,  1.35033800e+00,
        1.34852611e+00,  1.34791812e+00,  1.34655649e+00,  1.34444118e+00,
        1.34673590e+00,  1.33759979e+00,  1.34376097e+00,  1.34137536e+00,
        1.34101904e+00,  1.33836469e+00,  1.34312125e+00,  5.89386166e-01,
        5.90261268e-01,  5.91001619e-01,  5.92983371e-01,  5.93959106e-01,
        5.95705426e-01,  5.97060240e-01,  5.97919803e-01,  5.98748029e-01,
        6.00580835e-01,  6.02419530e-01,  6.04141444e-01,  6.09399496e-01,
        6.05195590e-01,  6.08130437e-01,  6.06163058e-01,  1.33467291e+00,
        1.33135861e+00,  1.33356642e+00,  6.06794009e-01,  1.33353622e+00,
        1.33089137e+00,  1.32688731e+00,  1.32761807e+00,  1.32974796e+00,
        1.32903147e+00,  1.32493231e+00,  1.32398795e+00,  1.32226357e+00,
        1.32287971e+00,  6.07745807e-01,  6.12412754e-01,  6.13657145e-01,
        6.15201677e-01,  6.15315692e-01,  6.30773304e-01,  6.31157494e-01,
        6.16854160e-01,  6.29631006e-01,  6.17432000e-01,  6.28063384e-01,
        6.26677455e-01,  6.19286055e-01,  6.19155567e-01,  6.26000554e-01,
        6.24703063e-01,  6.20784837e-01,  6.21955003e-01,  6.22970752e-01,
        6.23345569e-01,  1.31980662e+00,  1.31873099e+00,  1.31694076e+00,
        1.31427225e+00,  1.31682571e+00,  1.31850111e+00,  1.31540567e+00,
        1.31264081e+00,  1.30951757e+00,  1.31091189e+00,  1.31134978e+00,
        1.31199748e+00,  1.28373546e+00,  1.28437195e+00,  1.30686024e+00,
        1.30758594e+00,  1.30542684e+00,  1.28514574e+00,  1.30382561e+00,
        1.30288162e+00,  1.28719307e+00,  1.30136724e+00,  1.28803298e+00,
        1.29840484e+00,  1.29365910e+00,  1.29200739e+00,  1.28901556e+00,
        1.30090213e+00,  1.28924255e+00,  1.29646412e+00,  1.28976842e+00,
        1.29466676e+00,  1.29231633e+00,  1.29574206e+00,  1.29799070e+00,
        1.29958428e+00,  6.33692333e-01,  6.33526526e-01,  6.36668778e-01,
        1.27737563e+00,  1.27833574e+00,  1.28191198e+00,  1.27947509e+00,
        1.27980733e+00,  6.39340556e-01,  6.40891240e-01,  6.40003085e-01,
        6.43931497e-01,  6.44824140e-01,  6.45942076e-01,  6.38230678e-01,
        6.38041969e-01,  6.42578924e-01,  1.26944987e+00,  1.27080942e+00,
        1.27414338e+00,  1.27497234e+00,  1.27620437e+00,  1.27207000e+00,
        1.27257572e+00,  6.47740396e-01,  6.46910912e-01,  6.49333625e-01,
        6.50087115e-01,  6.52318365e-01,  6.52959112e-01,  6.54576429e-01,
        6.55215186e-01,  6.64085930e-01,  6.56454904e-01,  6.61610402e-01,
        6.59666762e-01,  6.57371358e-01,  6.62900038e-01,  6.57955638e-01,
        6.58706226e-01,  1.26811588e+00,  1.26878733e+00,  1.26673857e+00,
        1.26048628e+00,  1.26560294e+00,  1.26470672e+00,  1.26420761e+00,
        1.26340327e+00,  1.25912400e+00,  1.25793615e+00,  1.25680929e+00,
        6.65616378e-01,  6.68260383e-01,  6.67620678e-01,  6.70575768e-01,
        6.66993282e-01,  6.67059473e-01,  6.71684325e-01,  6.83115379e-01,
        6.72642906e-01,  6.82111187e-01,  6.80358246e-01,  6.79804833e-01,
        6.73510066e-01,  6.77890709e-01,  6.75950273e-01,  6.75209362e-01,
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        1.15052578e+00,  1.15145968e+00,  1.15286692e+00,  1.15601214e+00,
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        1.14697227e+00,  1.14815175e+00,  1.14789819e+00,  7.49153489e-01,
        7.51992222e-01,  7.54783274e-01,  7.53308879e-01,  7.51325287e-01,
        7.56718086e-01,  7.61247600e-01,  7.59024267e-01,  7.58002797e-01,
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        1.11316940e+00,  1.11347310e+00,  1.12679555e+00,  1.12379688e+00,
        1.12052578e+00,  1.12543896e+00,  1.11679129e+00,  1.11854678e+00,
        1.11932918e+00,  1.11647880e+00,  1.12288673e+00,  1.11531134e+00,
        1.11937384e+00,  1.12563409e+00,  7.62380745e-01,  7.68050425e-01,
        7.66897839e-01,  7.65950723e-01,  7.65275532e-01,  7.60917329e-01,
        7.63937722e-01,  7.64734607e-01,  1.11032815e+00,  7.70905749e-01,
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        7.75396105e-01,  7.76947720e-01,  7.77238021e-01,  7.80625720e-01,
        7.81056755e-01,  1.10630661e+00,  1.10519834e+00,  1.09958489e+00,
        1.10208341e+00,  1.10919413e+00,  1.10776253e+00,  1.09722104e+00,
        1.10039305e+00,  1.10273177e+00,  1.09851448e+00,  1.10976593e+00,
        1.10820223e+00,  7.83479770e-01,  7.82319426e-01,  7.82191150e-01,
        7.84591177e-01,  7.99687583e-01,  7.85341264e-01,  7.86745501e-01,
        7.97855779e-01,  7.96298647e-01,  7.95946946e-01,  7.95389609e-01,
        7.87401410e-01,  7.93354135e-01,  7.92364279e-01,  7.89976979e-01,
        7.91807064e-01,  7.89178668e-01,  7.91211265e-01,  1.09527316e+00,
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        1.09340372e+00,  1.09233394e+00,  1.09002744e+00,  1.08711981e+00,
        1.08903417e+00,  1.08856517e+00,  1.08628866e+00,  1.06737357e+00,
        1.06766066e+00,  1.08529852e+00,  1.06975392e+00,  1.06940531e+00,
        1.08273866e+00,  1.07292634e+00,  1.08013231e+00,  1.07372574e+00,
        1.08353207e+00,  1.08057783e+00,  1.08107430e+00,  1.07544553e+00,
        1.07493431e+00,  1.07805228e+00,  1.07676645e+00,  1.07852118e+00,
        8.02564047e-01,  8.00937501e-01,  8.01526828e-01,  1.06697749e+00,
        8.07522152e-01,  8.04652124e-01,  8.05958572e-01,  8.04102576e-01,
        8.08665189e-01,  8.09582928e-01,  8.12518033e-01,  8.11554485e-01,
        1.06485244e+00,  1.06315067e+00,  1.06144437e+00,  1.06175110e+00,
        1.05986310e+00,  1.05837806e+00,  8.13964893e-01,  8.18261030e-01,
        8.15105390e-01,  8.13856067e-01,  8.19751580e-01,  8.20706606e-01,
        8.15960148e-01,  8.17064432e-01,  8.16951568e-01,  8.22459810e-01,
        8.32086522e-01,  8.24785415e-01,  8.25369943e-01,  8.26334454e-01,
        8.28740204e-01,  8.27737865e-01,  8.29569406e-01,  8.27384777e-01,
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        1.03611128e+00,  1.03643306e+00,  1.03453711e+00,  8.36016761e-01,
        8.33783194e-01,  1.01533076e+00,  1.01570684e+00,  1.01721427e+00,
        1.01888929e+00,  1.02016447e+00,  1.01998198e+00,  8.42329151e-01,
        8.43986093e-01,  8.45366855e-01,  8.46098276e-01,  8.35504301e-01,
        8.38139098e-01,  8.46561801e-01,  8.40875530e-01,  8.38599164e-01,
        8.38427002e-01,  8.40619522e-01,  1.00970665e+00,  1.01074179e+00,
        1.00765823e+00,  1.01205203e+00,  1.01350003e+00,  1.01366812e+00,
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        8.55498597e-01,  8.54479327e-01,  8.51862509e-01,  8.56444892e-01,
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        9.64839534e-01,  9.66403982e-01,  9.66506494e-01,  8.95245047e-01,
        8.99135073e-01,  8.98756740e-01,  8.90292667e-01,  9.64191506e-01,
        9.08267538e-01,  8.95504401e-01,  9.01519347e-01,  9.62359136e-01,
        9.58938418e-01,  9.58109698e-01,  9.60176251e-01,  9.61686488e-01,
        9.10300590e-01,  9.06662723e-01,  9.60655535e-01,  9.55892442e-01,
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        9.12112826e-01,  9.17299029e-01,  9.51649316e-01,  9.26529840e-01,
        9.21972140e-01,  9.26966097e-01,  9.27914546e-01,  9.22430593e-01,
        9.23435263e-01,  9.02529026e-01,  9.16098910e-01,  9.24316263e-01,
        9.52945091e-01,  9.50064750e-01,  9.31063470e-01,  9.35733937e-01,
        9.15253758e-01,  9.04220635e-01,  9.35031620e-01,  9.47191965e-01,
        9.48302948e-01,  9.33158933e-01,  9.38453774e-01,  9.36808010e-01,
        9.44294058e-01,  9.41044664e-01,  9.41940171e-01,  9.38854211e-01,
        9.03716439e-01,  9.45496247e-01,  9.18357885e-01,  9.13077895e-01,
        9.43067127e-01,  9.14683326e-01,  9.13868047e-01,  9.02310589e-01,
        9.04740333e-01,  9.29166367e-01,  9.29277689e-01,  9.42409799e-01,
        9.33718703e-01,  9.54761472e-01,  9.53383434e-01,  9.45792861e-01,
        9.48801656e-01,  8.72911003e-01,  9.77668949e-01,  9.31713830e-01])

1
2


# 最大特征值
max(e)

1.5924617911775922

1
2


# 最小特征值
min(e)

-3.5387952353433675e-16

特征值分布直方图

 1
 2
 3
 4
 5
 6
 7
 8
 9
10


plt.figure(figsize=(12,8))

plt.hist(e, bins=100)
plt.xlim(0, 2)  # eigenvalues between 0 and 2

plt.title('Eigenvalue Histogram', fontsize=20)
plt.ylabel('Frequency', fontsize=25)
plt.xlabel('Eigenvalue', fontsize=25)
plt.tick_params(labelsize=20) # 设置坐标文字大小
plt.show()

目录

NetworkX实战

北京上海地铁站图数据挖掘

导入工具包

可视化辅助函数

字典按值排序辅助函数

导入地铁站连接表

创建图

从连接表创建图

检查是否导入成功

可视化

Shortest Path 最短路径

地铁导航系统

Node Degree

Degree Centrality

Eigenvector Centrality（可能不收敛）

Betweenness Centrality（必经之地）

Closeness Centrality（去哪儿都近）

PageRank

Katz Centrality

HITS Hubs and Authorities

NetworkX文档：社群属性 Clustering

三角形个数

Clustering Coefficient

计算全图Graphlet个数

导入工具包

导入全图

指定Graphlet

匹配Graphlet，统计个数

拉普拉斯矩阵特征值分解

导入工具包

创建图

邻接矩阵（Adjacency Matrix）

拉普拉斯矩阵（Laplacian Matrix）

归一化拉普拉斯矩阵（Normalized Laplacian Matrix）

特征值分解

特征值分布直方图