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UNIST Mathematical Sciences Network analysis with networkX(python library)
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Kyunghoon Kim
Network Analysis with networkX
Fundamentals of network theory-1
2014. 05. 14.
UNIST Mathematical Sciences
Kyunghoon Kim ( [email protected] )
5/14/2014 Fundamentals of network theory-1 1
Kyunghoon Kim
β A NETWORK is, in its simplest form, a collection of points joined together in pairs by lines.
What is a Network?
Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
5/14/2014 Fundamentals of network theory-1 2
Kyunghoon Kim
β A NETWORK is, in its simplest form, a collection of points joined together in pairs by lines.
β Network is general, but powerful means of representing patterns of connections or interactions between the parts of a system.
What is a Network?
Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
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Kyunghoon Kim
What is a Network?
Shipping (sea) networksTelecommunications networks
Air traffic networkData networks
Source: Britain From Above (http://www.bbc.co.uk/britainfromabove)
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Kyunghoon Kim
β Network thinking can be applied almost anywhere!
What is a Network?
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Kyunghoon Kim
β Network thinking can be applied almost anywhere!
Example of Network Thinking
http://journals.uic.edu/ojs/index.php/fm/article/view/941/863
My data sources were publicly released information reported in major newspapers such as the New York Times, Wall Street Journal, Washington Post, and the Los Angeles Times. As I monitored the investigation, it was apparent that the investigators would not be releasing all pertinent network/relationship information and actually may be releasing misinformation to fool the enemy. I soon realized that the data was not going to be as complete and accurate as I had grown accustomed to in mapping and measuring organizational networks.
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Example of Network Thinking
http://www.orgnet.com/hijackers.html
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Kyunghoon Kim
Example of Network Thinking
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Example of Network Thinking
Six (6) shortcuts were added to the network temporarily in order to collaborate and coordinate.These shortcuts reduced the average path length in the network by over 40% thus improving the information flow in the network - see Table 1.
When the network is brought closer together by these shortcuts, all of the pilots ended up in a small clique - the perfect structure to efficiently coordinate tasks and activities.
There is a constant dynamicbetween keeping the network hidden and actively using it to accomplish objectives (Baker and Faulkner, 1993).
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Kyunghoon Kim
Example of Network Thinking
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Kyunghoon Kim
Example of Network Thinking
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Kyunghoon Kim
Example of Network Thinking
Saddam Hussein
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Kyunghoon Kim
Example of Network Thinking
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Kyunghoon Kim
Example of Network Thinking
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An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 15Whereβs it applied?
Epidemiology (i.e. spread of diseases)e.g. spread of foot & mouth disease in the UK in 2001 over 75 days
URL: http://www.youtube.com/watch?v=PufTeIBNRJ4
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 16
Physicse.g. particle interactions, the structure of the universe
Whereβs it applied?
URL: http://www.youtube.com/watch?v=8C_dnP2fvxk
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 17
Engineeringe.g. creation of robust infrastructure (e.g. electricity, telecoms), rust formation (natural growth processes similar to diffusion limited aggregation)
Whereβs it applied?
URL: http://www.youtube.com/watch?v=lRZ2iEHFgGo URL: http://www.youtube.com/watch?v=AEoP-XtJueo
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 18
Technologye.g. mapping the online world, making networks resilient in the face of cyber-terrorism, optimising cellular networks, controlling air traffic
Whereβs it applied?
URL: http://www.youtube.com/watch?v=l-RoDv7c5ok URL: http://www.youtube.com/watch?v=o4g930pm8Ms
Vid not working
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 19Whereβs it applied?
Biologye.g. fish swimming in schools, ant colonies, birds flying in formation, crickets chirping in unison, giant honeybees shimmering
URL: http://www.youtube.com/watch?v=Sp8tLPDMUygURL: http://www.youtube.com/watch?v=YadP3w7vkJA
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 20
Medicinee.g. cell formation, nervous system, neural networks
Whereβs it applied?
Source: The Human Brain Book by Rita Carter
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 | 21
And, most interestinglyβ¦societye.g. interactions between people (e.g. Facebook; group behaviour)
Whereβs it applied?
URL: http://www.youtube.com/watch?v=9n9irapdON4 URL: http://www.youtube.com/watch?v=sD2yosZ9qDw
Kyunghoon Kim
β A network β also called a graph in the mathematical literature β is, as we have said, a collection of vertices joined by edges.
Networks and their Representation
Area Object Relation
Graph Theory Vertices Edges
Computer Science Nodes Links
Physics Sites Bonds
Sociology Actors Ties
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Networks and their Representation
Figure 6.1: Two small networks. (a) A simple graph, i.e., one having no multiedges or self-edges.(b) A network with both multiedges and self-edges.
Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
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Networks and their Representation
π΄π΄ππππ = οΏ½10
ππππ π‘π‘π‘π‘π‘π‘π‘π‘π‘ ππππ ππππ π‘π‘πππππ‘π‘ πππ‘π‘π‘π‘πππ‘π‘π‘π‘ππ π£π£π‘π‘π‘π‘π‘π‘πππ£π£π‘π‘ππ ππ ππππππ ππ,πππ‘π‘π‘π‘π‘π‘π‘πππππππ‘π‘.
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Kyunghoon Kim
β import networkx as nxG = nx.Graph()G.add_node(1)G.add_nodes_from([1,2,3,4,5])G.nodes()
β >>> [1, 2, 3, 4, 5]
Networks and their Representation
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β G.add_edge(1,2)G.add_edges_from([(1,2),(2,4),(2,5),(3,4)])G.edges()
β >>> [(1, 2), (2, 4), (2, 5), (3, 4)]
Networks and their Representation
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β nx.to_numpy_matrix(G)
β matrix([[ 0., 1., 0., 0., 0.],
[ 1., 0., 0., 1., 1.],
[ 0., 0., 0., 1., 0.],
[ 0., 1., 1., 0., 0.],
[ 0., 1., 0., 0., 0.]])
nx.draw(G)
Networks and their Representation
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Networks and their Representation
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Networks and their Representation
= π΄π΄ =
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Networks and their Representation
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Degree
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Degree
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Degree
ππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
ππ2 = οΏ½ππ=1
5
π΄π΄1ππ = 3
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Degree
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Degree
ππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
2ππ = οΏ½ππ=1
ππ
ππππ
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Kyunghoon Kim
Total Degree m
ππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
2ππ = οΏ½ππ=1
ππ
ππππ
ππ =12οΏ½ππ=1
ππ
ππππ =12οΏ½
ππππ
π΄π΄ππππ
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Kyunghoon Kim
Total Degree m
ππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
2ππ = οΏ½ππ=1
ππ
ππππ
ππ =12οΏ½ππ=1
ππ
ππππ =12οΏ½
ππππ
π΄π΄ππππππ
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Mean Degree c
ππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
2ππ = οΏ½ππ=1
ππ
ππππ
π£π£ =1πποΏ½ππ=1
ππ
ππππ =2ππππ
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Mean Degree c
π£π£ =1πποΏ½ππ=1
ππ
ππππ =2ππππ
>>> G.degree()
{1: 1, 2: 3, 3: 1, 4: 2, 5: 1}
>>> G.degree(2)
3
>>> Degree = G.degree().values()
[1, 3, 1, 2, 1]
>>> sum(Degree)/len(G.nodes())
1.6000000000000001
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Density or connectance of a graph
ππ2 =
12ππ(ππ β 1)
in a simple graph(i.e., one with no multiedgesor self-edges)
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Density or connectance of a graph
ππ =ππππ2
=2ππ
ππ(ππ β 1)
=π£π£
ππ β 1
ππ2 =
12ππ(ππ β 1)
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Kyunghoon Kim
Density or connectance of a graph
ππ =ππππ2
=2ππ
ππ(ππ β 1)
=π£π£
ππ β 1
>>> total = 0.5*5*(5-1)
>>> 4/total
0.4
>>> nx.density(G)
0.4
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β The in-degree is the # of ingoing edges connected to a vertex.
β The out-degree is the # of outgoing edges connected to a vertex.
β π΄π΄ππππ = 1 if there is an edge from ππ to ππ.
In-degree and out-degree
ππππππππ = οΏ½ππ=1
ππ
π΄π΄ππππ ππππππππππ = οΏ½ππ=1
ππ
π΄π΄ππππ
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β A path in a network is any sequence of vertices such that every consecutive pair of vertices in the sequence is connected by an edge in the network.
Paths
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β A path in a network is any sequence of vertices such that every consecutive pair of vertices in the sequence is connected by an edge in the network.
β A path is a route across the network that runs from vertex to vertex along the edges of the network.
Paths
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β The length of a path in a network is the # of edges traversed along the path (not the # of vertices).
Length of Paths
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β The length of a path in a network is the # of edges traversed along the path (not the # of vertices).
β π΄π΄ππππ = 1 if there is an edge from ππ to ππ,and 0 otherwise.
Length of Paths
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Kyunghoon Kim
β The length of a path in a network is the # of edges traversed along the path (not the # of vertices).
β π΄π΄ππππ = 1 if there is an edge from ππ to ππ,and 0 otherwise.
Length of Paths
ππππππ(2) = οΏ½
ππ=1
ππ
π΄π΄πππππ΄π΄ππππ = ππ2 ππππ
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β The length of a path in a network is the # of edges traversed along the path (not the # of vertices).
β π΄π΄ππππ = 1 if there is an edge from ππ to ππ,and 0 otherwise.
i β k , k βj
Length of Paths
ππππππ(2) = οΏ½
ππ=1
ππ
π΄π΄πππππ΄π΄ππππ = ππ2 ππππ
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β The length of a path in a network is the # of edges traversed along the path (not the # of vertices).
β π΄π΄ππππ = 1 if there is an edge from ππ to ππ,and 0 otherwise.
Length of Paths
ππππππ(2) = οΏ½
ππ=1
ππ
π΄π΄πππππ΄π΄ππππ = ππ2 ππππ
ππππππ(3) = οΏ½
ππ,ππ=1
π΄π΄πππππ΄π΄πππππ΄π΄ππππ = ππ3 ππππ
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β import numpy as npmatrix = nx.to_numpy_matrix(G)matrix*np.transpose(matrix)
matrix([[ 1., 0., 0., 1., 1.],
[ 0., 3., 1., 0., 0.],
[ 0., 1., 1., 0., 0.],
[ 1., 0., 0., 2., 1.],
[ 1., 0., 0., 1., 1.]])
Length of Paths
ππππππ(2) = οΏ½
ππ=1
ππ
π΄π΄πππππ΄π΄ππππ = ππ2 ππππ
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Loops
ππππππ(2) =
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β A shortest path is a path between two vertices such that no shorter path exists:
Shortest path (Geodesic path)
Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
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β A shortest path is a path between two vertices such that no shorter path exists:
β Shortest paths are necessarily self-avoiding.
β Shortest paths are not necessarily unique.
Shortest path (Geodesic path)
Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
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β The shortest distance between vertices ππ and ππis the smallest value of π‘π‘ such that ππππ ππππ > 0.
Shortest distance (Geodesic distance)
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β The shortest distance between vertices ππ and ππis the smallest value of π‘π‘ such that ππππ ππππ > 0.
β However, in practical case we use Dijkstraβsalgorithm. We will study it later.
Shortest distance (Geodesic distance)
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β The diameter of a graph is the length of the longest geodesic path between any pair of vertices in the network for which a path actually exists.
Diameter of a graph
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β Which are the most importantor central vertices in a network?
Measures and Metrics
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β Which are the most importantor central vertices in a network?
β Degree centrality,Closeness centrality,Betweenness centrality,Eigenvector centrality,Pagerank,Similarityβ¦
Measures and Metrics
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β Degree is sometimes called degree centrality in the social networks literature.
β In directed networks, vertices have both an in-degree and an out-degree, and both may be useful as measures of centrality in the appropriate circumstances.
Degree Centrality
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Degree Centrality
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Clustering Coefficient (local)
β For a vertex ππ, we define :
πΆπΆππ =(πππππππππ‘π‘π‘π‘ ππππ πππππππ‘π‘ππ ππππ πππ‘π‘πππππ‘πππππ‘π‘ππ ππππ ππ π‘π‘π‘πππ‘π‘ πππ‘π‘π‘π‘ π£π£πππππππ‘π‘π£π£π‘π‘π‘π‘ππ)
(πππππππππ‘π‘π‘π‘ ππππ πππππππ‘π‘ππ ππππ πππ‘π‘πππππ‘πππππ‘π‘ππ ππππ ππ)
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Clustering Coefficient (local)
β >>> nx.clustering(G)
β {1: 0.0, 2: 0.3333, 3: 0.6666, 4: 1.0, 5: 1.0}
πΆπΆ2 =242
=13
= 0.3333
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Clustering Coefficient (local)
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βUnclusteredβ network βClusteredβ networkNone of Egoβs friends know each other* All of Egoβs friends know each other
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
Kyunghoon Kim
Clustering Coefficient (local)
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β A real-world example: CEOs of Fortune 500 companiesβ’ Which companies share directors? Clusters are colour-coded
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
Kyunghoon Kim
β This measures the mean distance from a vertex to other vertices.
β Suppose ππππππ is the length of a geodesic path from ππ to ππ, meaning the # of edges along the path.
Closeness Centrality
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β This measures the mean distance from a vertex to other vertices.
β Suppose ππππππ is the length of a geodesic path from ππ to ππ, meaning the # of edges along the path.
β The mean shortest distance from ππ to ππ, averaged over all vertices ππ in the network, is
Closeness Centrality
ππππ =1πποΏ½
ππ
ππππππ
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β Exclude for ππ = ππ,
Closeness Centrality
ππππ =1
ππ β 1 οΏ½ππ(β ππ)
ππππππ
ππππ =1πποΏ½
ππ
ππππππ
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β This quantity takes low values for vertices that are separated from others by only a short shortest distance on average.
β which is the opposite of other measures.
β So commonly we take inverse :
Closeness Centrality
πΆπΆππ =1ππππ
=ππ
βππ ππππππ
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>>> nx.closeness_centrality(G)
{1: 0.5714, 2: 1.0, 3: 0.8, 4: 0.6666, 5: 0.6666}
Closeness Centrality
πΆπΆππ =1ππππ
=ππ
βππ ππππππ
πΆπΆ1 =4
βππ ππ1ππ=
41 + 2 + 2 + 2
= 0.5714
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β This measures the extent to which a vertex lies on paths between other vertices.
Betweenness Centrality
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β This measures the extent to which a vertex lies on paths between other vertices.
β For example, assume that every pair of vertices exchanges a message with equal probability per unit time and that messages always take the shortest path.
Betweenness Centrality
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β For example, assume that every pair of vertices exchanges a message with equal probability per unit time and that messages always take the shortest path.
β Question : if we wait a suitably long time until many messages have passed between each pair of vertices, how many messages, on average, will have passed through each vertex en route to their destination?
Betweenness Centrality
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β Question : if we wait a suitably long time until many messages have passed between each pair of vertices, how many messages, on average, will have passed through each vertex en route to their destination?
β Answer : since messages are passing down each shortest path at the same rate, the number passing through each vertex is simply proportional to the # of shortest paths the vertex lies on.
Betweenness Centrality
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β Let πππ π ππππ be 1 if vertex ππ lies on the shortest path from ππ to π‘π‘ and 0 if it does not or if there is no such path.
β The betweenness centrality π₯π₯ππ is given by
Betweenness Centrality
π₯π₯ππ = οΏ½π π ππ
πππ π ππππ
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β Note that the shortest paths between a pair of vertices need not be vertex independent, meaning they may pass through some of the same vertices.
β Define πππ π ππ to be the total # of shortest paths from ππ to π‘π‘ .
where πππ π π π ππ
πππ π π π = 0 if both πππ π ππππ and πππ π ππ are zero.
Betweenness Centrality
π₯π₯ππ = οΏ½π π ππ
πππ π ππππ
πππ π ππ
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>>> nx.betweenness_centrality(G)
{1: 0.0, 2: 0.5833, 3: 0.0833, 4: 0.0, 5: 0.0}
Betweenness Centrality
π₯π₯ππ = οΏ½π π ππ
πππ π ππππ
πππ π ππ
π₯π₯1 = οΏ½π π ππ
πππ π ππ1
πππ π ππ
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>>> nx.betweenness_centrality(G)
{1: 0.0, 2: 0.5833, 3: 0.0833, 4: 0.0, 5: 0.0}g = nx.Graph()
g.add_edges_from([(1,2),(2,3),(2,4),(2,5),(3,4),(3,5)])end = 6
for i in range(1,end):
for j in range(1,end):
#print 'from', i, 'to', j
print([p for p in nx.all_shortest_paths(g,i,j) if len(p)>2])
#print nx.dijkstra_path(g,i,j)
#print '=============='
nx.betweenness_centrality(g, normalized=False)
plt.cla()
nx.draw(g)
Betweenness Centrality
π₯π₯2 = οΏ½π π ππ
πππ π ππ2
πππ π ππ= 3 +
12
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>>> nx.betweenness_centrality(G)
{1: 0.0, 2: 0.5833, 3: 0.0833, 4: 0.0, 5: 0.0}
[[1, 2, 3]]
[[1, 2, 4]]
[[1, 2, 5]]
[[3, 2, 1]]
[[4, 2, 1]]
[[4, 2, 5], [4, 3, 5]]
[[5, 2, 1]]
[[5, 2, 4], [5, 3, 4]]
Betweenness Centrality
π₯π₯2 = οΏ½π π ππ
πππ π ππ2
πππ π ππ= 3 +
12
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>>> nx.betweenness_centrality(G)
{1: 0.0, 2: 0.5833, 3: 0.0833, 4: 0.0, 5: 0.0}
To normalize,
we use the following equation,2
(ππβ1)(ππβ2)
Then,
Betweenness Centrality
π₯π₯2 =2
4 Γ 3οΏ½π π ππ
πππ π ππ2
πππ π ππ=
16
3 +12
= 0.5833
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Appendix
β G.add_edges_from([(1,3),(3,6),(3,7),(3,8),(6,8),(7,8),(8,9)])
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β nx.draw_spring(G)
β nx.to_numpy_matrix(G)
Appendix
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β Newman, Mark. Networks: an introduction. OUP Oxford, 2009.
β An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
β http://journals.uic.edu/ojs/index.php/fm/article/view/941/863
β http://www.orgnet.com/hijackers.html
Reference
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