Modularity and community structure in networks
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Modularity and community structure in networks The Proceedings of the National Academy of Sciences of the United States of America, June 2006 Author:M.E.J.Newman 1
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Modularity and community structure in networks. The Proceedings of the National Academy of Sciences of the United States of America, June 2006 Author:M.E.J.Newman. O utline. The definition of community The arguments of modularity and graph Modularity - PowerPoint PPT Presentation
Transcript of Modularity and community structure in networks
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Modularity and community structure in networks
The Proceedings of the National Academy of Sciences of the United States of America, June 2006
Author:M.E.J.Newman
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The definition of community The arguments of modularity
and graph Modularity Dividing network into two communities Dividing network into more than two
communities Experiment
Outline
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Community: the densely connected groups of vertices, with only sparser connections between groups.
The definition of community
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A:adjacency matrix ki: the degrees of vertex i. si= m: total edges of the graph n: total vertices in the graph
The arguments of modularity and graph
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Q=(number of edges within communities)-(expected number of such edges)
= = = = = (1)
Modularity
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Modularity Q1=0.3479
Modularity Q2=0.2043
The example of modularity
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(2) S= B:nn matrix , called modularity matrix
Dividing network into two communities
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Ex.
=(+++) = =
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m=14
A=
k1=3,k2=2,k3=3,k4=4,k5=4,k6=4,k7=4,k8=3,k9=1
Example
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B11= A11 – = 0 - = -0.3214
B12= A12 – = 1 - = 0.7857
mkk
211
289
mkk
221
286
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Let Ex = = = = : the ith eigenvector of B
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= = = = = (3) : the eigenvalue of B, >
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=
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Dividing network into more than two communities
∆𝑄=1
2𝑚 ¿
g
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= = =
=
Dividing network into more than two communities
333231232221131211
333332313,3222322212,2111312111,1 )()()(
BBBBBBBBB
ssBBBssBBBssBBB
ba if,0ba if,1
,ba
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Experiment
