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Modularity and community structure in networks

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

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Modularity and community structure in networks

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  1. 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

  2. Outline • The definition of community • The arguments of modularity and graph • Modularity • Dividing network into two communities • Dividing network into more than two communities • Experiment

  3. The definition of community • Community: the densely connected groups of vertices, with only sparser connections between groups.

  4. The arguments of modularity and graph • A:adjacency matrix • ki: the degrees of vertex i. • si= • m: total edges of the graph • n: total vertices in the graph

  5. Modularity • Q=(number of edges within communities)-(expected number of such edges) • = • = • = • = • = (1)

  6. The example of modularity Modularity Q1=0.3479 Modularity Q2=0.2043

  7. Dividing network into two communities • (2) • S= • B:nn matrix , called modularity matrix

  8. Ex. • =(+++) • = • =

  9. Example • m=14 A= • k1=3,k2=2,k3=3,k4=4,k5=4,k6=4,k7=4,k8=3,k9=1

  10. B11= A11 – = 0 - = -0.3214 B12= A12 – = 1 - = 0.7857

  11. Let • Ex • = • = • = • = • : the ith eigenvector of B

  12. = • = • = • = • = (3) • : the eigenvalue of B,>

  13. =

  14. Dividing network into more than two communities g

  15. Dividing network into more than two communities = = = =

  16. Experiment

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