1. Detecting Important Nodes to Community Structure
Ying Fan
Beijing Normal University
27/04/2011

2. Outline Community structure Centrality metric
Two kind of important nodes
Examples
Discussion

3. Community Structure Physics collaboration network
Palla et al. Nature 435, 9 (2005)

4. The significance of Community

5. Centrality Degree Betweenness Eigenvector Leverage centrality
Subgraph centrality …
To community?

6. The importance of node to community
The importance of node k to communities as the relative change in the c largest eigenvalues of the network adjacency matrix upon removal:

7. Index I
We focus on the relative change in the eigenvalues of the adjacency matrix upon the nodes removal.

8. 8

9. Differentiate nodes
There are always two kind of important nodes in community : “community core” and “bridge”
How to differentiate them? 9

10. How? If the network has just two communities… The cut size is where
One could regard the elements of of the graph Laplacian, with vertices whose corresponding elements are close to zero having nearly equal membership in both. 10

11. If the network has more communities…suppose it has c communities
After some deductions we obtain:
Let , we know 11

12. w-score

13. Illustration 13

14. GN Benchmark

15. LFR Benchmark
We analyze LFR benchmark which size is 1000, mixing parameter is 0.25, degree power law exponent is 2.5, community size power law exponent is 1.0.
no “bridge” node 15

16. Real world network -Zachary's karate club network
ZK club 16

17. Real world network -The Word Association Network17

18. Real world network -Scientists collaboration network

19. Real world network -C. Elegans neuronal network

20. In weighted networks -SFI network

21. Discussion
Limitation:
Weighted networks
Directed networks?

22. Thanks for listening!