In [1]:
import networkx as nx
from custom import custom_funcs as cf
from networkx.algorithms import bipartite
from circos import CircosPlot
import numpy as np
import matplotlib.pyplot as plt
%load_ext autoreload
%autoreload 2
%matplotlib inline
Bipartite graphs are graphs that have two (bi-) partitions (-partite) of nodes. Nodes within each partition are not allowed to be connected to one another; rather, they can only be connected to nodes in the other partition.
Bipartite graphs can be useful for modelling relations between two sets of entities. We will explore the construction and analysis of bipartite graphs here.
Let's load a crime data bipartite graph and quickly explore it.
This bipartite network contains persons who appeared in at least one crime case as either a suspect, a victim, a witness or both a suspect and victim at the same time. A left node represents a person and a right node represents a crime. An edge between two nodes shows that the left node was involved in the crime represented by the right node.
In [2]:
G = cf.load_crime_network()
G.edges(data=True)[0:5]
Out[2]:
[('p213', 'c262', {'role': 'Witness'}),
('p61', 'c112', {'role': 'Suspect'}),
('p386', 'c161', {'role': 'Suspect'}),
('c55', 'p405', {'role': 'Victim'}),
('c55', 'p404', {'role': 'Victim'})]
In [3]:
G.nodes(data=True)[0:10]
Out[3]:
[('p213', {'bipartite': 'person', 'gender': 1}),
('p61', {'bipartite': 'person', 'gender': 1}),
('p386', {'bipartite': 'person', 'gender': 1}),
('c55', {'bipartite': 'crime'}),
('c267', {'bipartite': 'crime'}),
('p805', {'bipartite': 'person', 'gender': 1}),
('p590', {'bipartite': 'person', 'gender': 1}),
('p329', {'bipartite': 'person', 'gender': 1}),
('c133', {'bipartite': 'crime'}),
('p616', {'bipartite': 'person', 'gender': 1})]
Find the bipartite projection function in the NetworkX bipartite
module docs, and use it to obtain the unipartite
projection of the bipartite graph.
In [4]:
person_nodes = [n for n in G.nodes() if G.node[n]['bipartite'] == 'person']
pG = bipartite.projection.projected_graph(G, person_nodes)
pG.nodes(data=True)[0:5]
Out[4]:
[('p213', {'bipartite': 'person', 'gender': 1}),
('p563', {'bipartite': 'person', 'gender': 1}),
('p133', {'bipartite': 'person', 'gender': 1}),
('p61', {'bipartite': 'person', 'gender': 1}),
('p381', {'bipartite': 'person', 'gender': 0})]
In [5]:
nodes = sorted(pG.nodes(), key=lambda x: (pG.node[x]['gender'], len(pG.neighbors(x))))
edges = pG.edges()
edgeprops = dict(alpha=0.1)
node_cmap = {0:'blue', 1:'red'}
nodecolor = [node_cmap[pG.node[n]['gender']] for n in nodes]
fig = plt.figure(figsize=(6,6))
ax = fig.add_subplot(111)
c = CircosPlot(nodes, edges, radius=10, ax=ax, fig=fig, edgeprops=edgeprops, nodecolor=nodecolor)
c.draw()
c.fig.savefig('images/crime-person.png', dpi=300)
In [6]:
crime_nodes = [n for n in G.nodes() if G.node[n]['bipartite'] == 'crime']
cG = bipartite.projection.projected_graph(G, crime_nodes)
In [7]:
nodes = sorted(cG.nodes(), key=lambda x: len(cG.neighbors(x)))
edges = cG.edges()
edgeprops = dict(alpha=0.1)
nodecolor = plt.cm.viridis(np.arange(len(nodes)) / len(nodes))
fig = plt.figure(figsize=(6,6))
ax = fig.add_subplot(111)
c = CircosPlot(nodes, edges, radius=10, ax=ax, fig=fig, edgeprops=edgeprops, nodecolor=nodecolor)
c.draw()
plt.savefig('images/crime-crime.png', dpi=300)
NetworkX also implements centrality measures for bipartite graphs, which allows you to obtain their metrics without first converting to a particular projection. This is useful for exploratory data analysis.
Try the following challenges, referring to the API documentation to help you:
In [8]:
# Degree Centrality
bpdc = bipartite.degree_centrality(G, person_nodes)
sorted(bpdc.items(), key=lambda x: x[1], reverse=True)
Out[8]:
[('p815', 0.045372050816696916),
('p2', 0.039927404718693285),
('p425', 0.032667876588021776),
('p220', 0.0308529945553539),
('p56', 0.025408348457350273),
('p715', 0.021778584392014518),
('c110', 0.02171290711700844),
('p356', 0.019963702359346643),
('p695', 0.019963702359346643),
('p514', 0.019963702359346643),
('p531', 0.018148820326678763),
('p404', 0.018148820326678763),
('c153', 0.0180940892641737),
('p336', 0.016333938294010888),
('p659', 0.016333938294010888),
('p592', 0.016333938294010888),
('p497', 0.016333938294010888),
('p696', 0.016333938294010888),
('p474', 0.016333938294010888),
('p115', 0.016333938294010888),
('c43', 0.015681544028950542),
('c14', 0.015681544028950542),
('c95', 0.015681544028950542),
('p37', 0.014519056261343012),
('p146', 0.014519056261343012),
('p88', 0.014519056261343012),
('p10', 0.014519056261343012),
('p729', 0.014519056261343012),
('c39', 0.01447527141133896),
('c161', 0.013268998793727381),
('p414', 0.012704174228675136),
('p277', 0.012704174228675136),
('p549', 0.012704174228675136),
('p291', 0.012704174228675136),
('p74', 0.012704174228675136),
('p214', 0.012704174228675136),
('p797', 0.012704174228675136),
('p690', 0.012704174228675136),
('p159', 0.012704174228675136),
('p572', 0.012704174228675136),
('p384', 0.010889292196007259),
('p533', 0.010889292196007259),
('p51', 0.010889292196007259),
('p62', 0.010889292196007259),
('p381', 0.010889292196007259),
('p767', 0.010889292196007259),
('p128', 0.010889292196007259),
('p108', 0.010889292196007259),
('p79', 0.010889292196007259),
('p736', 0.010889292196007259),
('p190', 0.010889292196007259),
('p413', 0.010889292196007259),
('c187', 0.01085645355850422),
('c78', 0.01085645355850422),
('c36', 0.01085645355850422),
('c189', 0.009650180940892641),
('c24', 0.009650180940892641),
('c97', 0.009650180940892641),
('c225', 0.009650180940892641),
('c46', 0.009650180940892641),
('c112', 0.009650180940892641),
('c19', 0.009650180940892641),
('c28', 0.009650180940892641),
('p129', 0.009074410163339382),
('p577', 0.009074410163339382),
('p642', 0.009074410163339382),
('p155', 0.009074410163339382),
('p681', 0.009074410163339382),
('p502', 0.009074410163339382),
('p17', 0.009074410163339382),
('p536', 0.009074410163339382),
('p788', 0.009074410163339382),
('p811', 0.009074410163339382),
('p242', 0.009074410163339382),
('p344', 0.009074410163339382),
('p396', 0.009074410163339382),
('c262', 0.008443908323281062),
('c206', 0.008443908323281062),
('c87', 0.008443908323281062),
('c442', 0.008443908323281062),
('c50', 0.008443908323281062),
('c290', 0.008443908323281062),
('c152', 0.008443908323281062),
('c60', 0.008443908323281062),
('p67', 0.007259528130671506),
('p397', 0.007259528130671506),
('p812', 0.007259528130671506),
('p131', 0.007259528130671506),
('p600', 0.007259528130671506),
('p680', 0.007259528130671506),
('p54', 0.007259528130671506),
('p304', 0.007259528130671506),
('p1', 0.007259528130671506),
('p333', 0.007259528130671506),
('p515', 0.007259528130671506),
('p221', 0.007259528130671506),
('p87', 0.007259528130671506),
('p410', 0.007259528130671506),
('p405', 0.007259528130671506),
('p112', 0.007259528130671506),
('p343', 0.007259528130671506),
('p345', 0.007259528130671506),
('p351', 0.007259528130671506),
('p174', 0.007259528130671506),
('p763', 0.007259528130671506),
('c56', 0.00723763570566948),
('c155', 0.00723763570566948),
('c29', 0.00723763570566948),
('c413', 0.00723763570566948),
('c251', 0.00723763570566948),
('c114', 0.00723763570566948),
('c435', 0.00723763570566948),
('c72', 0.00723763570566948),
('c47', 0.00723763570566948),
('c169', 0.00723763570566948),
('c62', 0.00723763570566948),
('c86', 0.006031363088057901),
('c480', 0.006031363088057901),
('c94', 0.006031363088057901),
('c35', 0.006031363088057901),
('c52', 0.006031363088057901),
('c405', 0.006031363088057901),
('c271', 0.006031363088057901),
('c248', 0.006031363088057901),
('c432', 0.006031363088057901),
('c79', 0.006031363088057901),
('c309', 0.006031363088057901),
('c257', 0.006031363088057901),
('c116', 0.006031363088057901),
('c77', 0.006031363088057901),
('c204', 0.006031363088057901),
('c23', 0.006031363088057901),
('c61', 0.006031363088057901),
('c332', 0.006031363088057901),
('c263', 0.006031363088057901),
('c379', 0.006031363088057901),
('c104', 0.006031363088057901),
('c300', 0.006031363088057901),
('c22', 0.006031363088057901),
('c138', 0.006031363088057901),
('c63', 0.006031363088057901),
('c80', 0.006031363088057901),
('c82', 0.006031363088057901),
('c181', 0.006031363088057901),
('c67', 0.006031363088057901),
('p202', 0.0054446460980036296),
('p225', 0.0054446460980036296),
('p686', 0.0054446460980036296),
('p411', 0.0054446460980036296),
('p818', 0.0054446460980036296),
('p352', 0.0054446460980036296),
('p452', 0.0054446460980036296),
('p663', 0.0054446460980036296),
('p439', 0.0054446460980036296),
('p57', 0.0054446460980036296),
('p543', 0.0054446460980036296),
('p245', 0.0054446460980036296),
('p219', 0.0054446460980036296),
('p746', 0.0054446460980036296),
('p809', 0.0054446460980036296),
('p406', 0.0054446460980036296),
('p216', 0.0054446460980036296),
('p82', 0.0054446460980036296),
('p584', 0.0054446460980036296),
('p303', 0.0054446460980036296),
('p106', 0.0054446460980036296),
('p89', 0.0054446460980036296),
('p723', 0.0054446460980036296),
('p472', 0.0054446460980036296),
('p613', 0.0054446460980036296),
('p466', 0.0054446460980036296),
('p370', 0.0054446460980036296),
('p719', 0.0054446460980036296),
('p363', 0.0054446460980036296),
('p362', 0.0054446460980036296),
('p152', 0.0054446460980036296),
('p392', 0.0054446460980036296),
('p764', 0.0054446460980036296),
('p512', 0.0054446460980036296),
('p567', 0.0054446460980036296),
('p12', 0.0054446460980036296),
('p636', 0.0054446460980036296),
('p740', 0.0054446460980036296),
('c151', 0.0048250904704463205),
('c160', 0.0048250904704463205),
('c40', 0.0048250904704463205),
('c30', 0.0048250904704463205),
('c42', 0.0048250904704463205),
('c227', 0.0048250904704463205),
('c166', 0.0048250904704463205),
('c75', 0.0048250904704463205),
('c212', 0.0048250904704463205),
('c438', 0.0048250904704463205),
('c59', 0.0048250904704463205),
('c245', 0.0048250904704463205),
('c90', 0.0048250904704463205),
('c388', 0.0048250904704463205),
('c140', 0.0048250904704463205),
('c315', 0.0048250904704463205),
('c236', 0.0048250904704463205),
('c119', 0.0048250904704463205),
('c173', 0.0048250904704463205),
('c296', 0.0048250904704463205),
('c232', 0.0048250904704463205),
('c217', 0.0048250904704463205),
('c44', 0.0048250904704463205),
('c81', 0.0048250904704463205),
('c417', 0.0048250904704463205),
('c238', 0.0048250904704463205),
('c320', 0.0048250904704463205),
('c65', 0.0048250904704463205),
('c31', 0.0048250904704463205),
('c425', 0.0048250904704463205),
('c370', 0.0048250904704463205),
('c111', 0.0048250904704463205),
('c26', 0.0048250904704463205),
('c293', 0.0048250904704463205),
('c378', 0.0048250904704463205),
('c117', 0.0048250904704463205),
('c218', 0.0048250904704463205),
('c419', 0.0048250904704463205),
('c107', 0.0048250904704463205),
('c344', 0.0048250904704463205),
('c11', 0.0048250904704463205),
('c126', 0.0048250904704463205),
('c316', 0.0048250904704463205),
('c301', 0.0048250904704463205),
('c33', 0.0048250904704463205),
('c51', 0.0048250904704463205),
('c297', 0.0048250904704463205),
('c108', 0.0048250904704463205),
('c515', 0.0048250904704463205),
('c294', 0.0048250904704463205),
('c57', 0.0048250904704463205),
('p662', 0.003629764065335753),
('p243', 0.003629764065335753),
('p97', 0.003629764065335753),
('p454', 0.003629764065335753),
('p313', 0.003629764065335753),
('p424', 0.003629764065335753),
('p490', 0.003629764065335753),
('p45', 0.003629764065335753),
('p41', 0.003629764065335753),
('p481', 0.003629764065335753),
('p162', 0.003629764065335753),
('p778', 0.003629764065335753),
('p776', 0.003629764065335753),
('p449', 0.003629764065335753),
('p676', 0.003629764065335753),
('p299', 0.003629764065335753),
('p437', 0.003629764065335753),
('p199', 0.003629764065335753),
('p400', 0.003629764065335753),
('p634', 0.003629764065335753),
('p249', 0.003629764065335753),
('p334', 0.003629764065335753),
('p524', 0.003629764065335753),
('p284', 0.003629764065335753),
('p775', 0.003629764065335753),
('p779', 0.003629764065335753),
('p335', 0.003629764065335753),
('p354', 0.003629764065335753),
('p117', 0.003629764065335753),
('p637', 0.003629764065335753),
('p739', 0.003629764065335753),
('p305', 0.003629764065335753),
('p338', 0.003629764065335753),
('p793', 0.003629764065335753),
('p420', 0.003629764065335753),
('p435', 0.003629764065335753),
('p692', 0.003629764065335753),
('p783', 0.003629764065335753),
('p184', 0.003629764065335753),
('p470', 0.003629764065335753),
('p58', 0.003629764065335753),
('p326', 0.003629764065335753),
('p33', 0.003629764065335753),
('p568', 0.003629764065335753),
('p744', 0.003629764065335753),
('p774', 0.003629764065335753),
('p315', 0.003629764065335753),
('p48', 0.003629764065335753),
('p114', 0.003629764065335753),
('p269', 0.003629764065335753),
('p412', 0.003629764065335753),
('p283', 0.003629764065335753),
('p46', 0.003629764065335753),
('p331', 0.003629764065335753),
('p821', 0.003629764065335753),
('p253', 0.003629764065335753),
('p666', 0.003629764065335753),
('p371', 0.003629764065335753),
('p297', 0.003629764065335753),
('p264', 0.003629764065335753),
('p101', 0.003629764065335753),
('p73', 0.003629764065335753),
('p652', 0.003629764065335753),
('p267', 0.003629764065335753),
('p415', 0.003629764065335753),
('p167', 0.003629764065335753),
('p357', 0.003629764065335753),
('p532', 0.003629764065335753),
('p633', 0.003629764065335753),
('p77', 0.003629764065335753),
('p40', 0.003629764065335753),
('p418', 0.003629764065335753),
('p702', 0.003629764065335753),
('p513', 0.003629764065335753),
('p441', 0.003629764065335753),
('p78', 0.003629764065335753),
('p398', 0.003629764065335753),
('p145', 0.003629764065335753),
('p561', 0.003629764065335753),
('p731', 0.003629764065335753),
('p171', 0.003629764065335753),
('p664', 0.003629764065335753),
('p361', 0.003629764065335753),
('p511', 0.003629764065335753),
('p649', 0.003629764065335753),
('p244', 0.003629764065335753),
('p365', 0.003629764065335753),
('p119', 0.003629764065335753),
('p563', 0.003629764065335753),
('p294', 0.003629764065335753),
('p247', 0.003629764065335753),
('p104', 0.003629764065335753),
('p758', 0.003629764065335753),
('p21', 0.003629764065335753),
('p476', 0.003629764065335753),
('c55', 0.00361881785283474),
('c130', 0.00361881785283474),
('c216', 0.00361881785283474),
('c416', 0.00361881785283474),
('c326', 0.00361881785283474),
('c235', 0.00361881785283474),
('c349', 0.00361881785283474),
('c318', 0.00361881785283474),
('c383', 0.00361881785283474),
('c361', 0.00361881785283474),
('c243', 0.00361881785283474),
('c185', 0.00361881785283474),
('c399', 0.00361881785283474),
('c313', 0.00361881785283474),
('c298', 0.00361881785283474),
('c219', 0.00361881785283474),
('c88', 0.00361881785283474),
('c91', 0.00361881785283474),
('c284', 0.00361881785283474),
('c207', 0.00361881785283474),
('c277', 0.00361881785283474),
('c341', 0.00361881785283474),
('c339', 0.00361881785283474),
('c53', 0.00361881785283474),
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('c122', 0.00361881785283474),
('c71', 0.00361881785283474),
('c282', 0.00361881785283474),
('c98', 0.00361881785283474),
('c241', 0.00361881785283474),
('c157', 0.00361881785283474),
('c352', 0.00361881785283474),
('c467', 0.00361881785283474),
('c324', 0.00361881785283474),
('c398', 0.00361881785283474),
('c196', 0.00361881785283474),
('c83', 0.00361881785283474),
('c221', 0.00361881785283474),
('c41', 0.00361881785283474),
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('c356', 0.00361881785283474),
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('c451', 0.00361881785283474),
('c396', 0.00361881785283474),
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('c73', 0.00361881785283474),
('c406', 0.00361881785283474),
('c172', 0.00361881785283474),
('c308', 0.00361881785283474),
('c156', 0.00361881785283474),
('c174', 0.00361881785283474),
('c58', 0.00361881785283474),
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('c165', 0.00361881785283474),
('c321', 0.00361881785283474),
('c380', 0.00361881785283474),
('c353', 0.00361881785283474),
('c452', 0.00361881785283474),
('c360', 0.00361881785283474),
('c162', 0.00361881785283474),
('c461', 0.00361881785283474),
('c234', 0.00361881785283474),
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('c211', 0.00361881785283474),
('c530', 0.00361881785283474),
('c322', 0.00361881785283474),
('c365', 0.00361881785283474),
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('c20', 0.00361881785283474),
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('c287', 0.00361881785283474),
('c190', 0.00361881785283474),
('c481', 0.00361881785283474),
('c266', 0.00361881785283474),
('c68', 0.00361881785283474),
('c199', 0.00361881785283474),
('c188', 0.00361881785283474),
('c254', 0.00361881785283474),
('c101', 0.00361881785283474),
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...]
Content source: SubhankarGhosh/NetworkX
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