Effects of External Links on the Synchronization of Community Networks

1. Effects of External Links on the Synchronization of Community Networks ZHAO Ming 2010.10

2. Collaborators: • ChangsongZhou周昌松， 香港浸会大学 • Jinhu Lv吕金虎，中科院数学与系统科学研究员 • Bambi Hu 胡斑比，香港浸会大学 • Choi Heng Lai 赖载兴，新加坡国立大学

3. Outline • Community and inter-link • Effects of external link number on network synchronization • Effects of external link connecting strategy on network synchronization • Effects of external link strength on network synchronization • Conclusion

4. Community and external link • Community: a group of nodes that have denser links among them than with the rest of network • External link: links that connect nodes between different communities • Internal link: links that connect nodes in the same community internal link External link community

5. Community Network Model Individual node:Rossler attractor with bounded synchronization region.

6. Effects of external link number Take two random networks as communities and rewire the internal link to the other community, after the first external link emerges, the community network has the largest modularity, with the increasing of rewiring probability, the modularity will decrease by and by. We investigate the synchronization performance of the community network as whole and that of the individual community.

7. Correlation VS. coupling strength The whole network: larger modularity will decrease the global synchronizability, and when there are few external links, the global synchronizability will not increase monotonically with the coupling strength. Individual community: fewer external links do not always mean better synchronization performance of community.

8. Dynamical modularity VS. coupling strength Definition:the correlation between each pair of nodes at some coupling strength is the strength of dynamical links between the two nodes, which will change with the vary of coupling strength. There are two peaks and clearly, larger topological modularity will ensure larger dynamical modularity.

9. Correlation and dynamical modularity VS. topological modularity Decrease monotonically Decrease first and then increase Increase monotonically

10. Cat cortical network 53 Nodes, 826 weighted links, 4 communities The network is rewired to decrease or increase the topological modularity

11. Correlation VS. topological modularity of Cat cortical network

12. Dynamical modularity VS. topological modularity of Cat cortical network

13. Effects of external link connecting strategy The connecting strategies, such as connecting different communities • by randomly selected nodes, • by hub nodes, • or by connecting hub nodes with randomly selected nodes in the other communities. will affect the synchronization performance.

14. Sketch map of connecting strategy Take BA networks as communities and arrange nodes in the same community according to degree, and then connect • I: random selected nodes • II: nodes selected with probability proportional to k^4 III: hub nodes and random selected nodes • IV: nodes with larger degree • V: nodes with middle degree • VI: nodes with smaller degree between different communities. III IV V VI

15. Correlation VS. coupling strength I: random selected nodes II: nodes selected with probability proportional to k^4 III: hub node and random selected nodes IV: nodes with larger degree V: nodes with middle degree VI: nodes with smaller degree Community: no great differences between different strategies, but with the increasing of external links, strategy VI has the largest average correlation Network: when the external links are too few the differences are not prominent, and when there are enough external links, the global synchronization performance will change from better to worse according to the sequence I>VI>III>IV>V>II.

16. Dynamical modularity VS. coupling strength

17. Effects of external link strength I The strength of internal links is kept 1, and the strength of external links changes from 0.5 to 3.

18. Effects of external link strength II

19. Conclusion of this work • External links can increase the global network synchronizability, but they also decrease the synchronization performance of individual community in some modularity region; • Connecting hub nodes does not guarantee better global network synchronizability; • Stronger strength of external links will enhance global network synchronizability, but narrow down the coupling strength region.

20. Thank you！