1. An Effective Method to Improve the Resistance to Frangibility in Scale-free Networks Kaihua Xu HuaZhong Normal University

2. Outline  Introduction  The Characters of Scale-Free networks  The Research on Improving the Resistance to Frangibility of Scale-Free Networks  Conclusion

3.  Introduction  More and more people start to research complex network in academic circle.  The common characteristic of complex networks in nature.  Based on the characteristic of complex networks, two models were proposed in this paper.

4.  Recently, three important complex network models, which are random graph model, small-world network and scale-free network model, have been put forward.  In these research results, it is always distinguished very clearly that the topologies of complex networks are belong to ER, WS or BA models.  The Characters of Scale-Free Networks

5.  The Characters of Scale-Free  Networks(2)  The robustness of scale-free networks to random failures. If the nodes which had few links were moved randomly, only a few nodes would decrease links, and the decreased links would not affect the structure of scale-free networks, so they were robust when nodes moved randomly.

6.  The Characters of Scale-Free  networks(3)  The frangibility of scale-free networks to spiteful attack. If the hub nodes were spitefully attacked, a lot of nodes which linked to the hub nodes would decrease links, and even some nodes would be isolated, the structure of the networks was destroyed, so the scale-free networks were frangible when nodes attacked spitefully.

7.  The Research on Improving the Resistance to Frangibility of Scale-Free Networks  The model research based on minimum degree to improving the resistance to frangibility of scale-free networks.  The model research based on Rank to improving the resistance to frangibility of scale-free networks.

8.  The model based on minimum degree In order to improving the resistance to frangibility, the hub nodes which was the key of frangibility should be decreased, the entropy of degree distribution could measure the heterogeneity of degree distribution in scale-free networks, if the entropy of degree distribution was optimized, the degree would be equally distributed, and the increase of hub nodes would be restrained.

9.  The model based on minimum degree(2) The solution to the model could be expressed as follows:

10. An example was given with N=10000,Figure 1 reflect the relationship between minimal degree and endurable probability for different. Figure 1 Endurable probability

11. Figure 2 showed the relationship between entropy of degree distribution and minimal degree for different. Entropy of degree distribution Figure 2 Entropy of degree distribution H

12. Figure 3 displayed the relationship between capability of resistance to frangibility and average degree for different. Capability of resistance to frangibility Figure 3 capability of resistance to frangibility

13.  The model based on Rank The way to measure off rank In most real complex networks, the structures were not equally important which were called “group character”. For example the links between different communities were few, but they were very important to compose the whole complex networks.

14. Figure 4 measure off rank From figure 4,Via the measure off rank in the paper, the key nodes and bridge nodes which linked to different communities in complex networks would be found. Figure 1 showed a small networks with nodes N=10, according to above method, the networks was divided into two ranks, the nodes in the first rank were:1,2,5, and the nodes in the second rank were: 3,4,6,7,8,9,10.

15.  The model based on Rank(2) Topology partition of scale-free networks In networks, adding or removing an edge was a partition. Partition changed the topology structure of complex networks. If we partitioned and reconstructed properly, the topology of complex networks would be optimized.

16.  The model based on Rank(3) Topology optimization of scale-free networks The progress of building complex networks was complex, but we could do something to control and optimize it when the complex networks were being generated. In order to optimize the topology of complex networks we could amend its connections, made sure it was optimally connected in local, so that the topology of the whole complex networks could be optimized.

17. An example was given with N=10000,Figure 5 reflect the relationship between the degree k and the degree distribution p(k). Figure 5 degree distribution

18. From figure 5 we analyzed the degree distribution of the complex networks; it obeyed function.On the top right corner of figure 6 showed the power-law function. below showed the degree distribution of the complex networks. They were similar; that was to say measure off rank, partition and connection optimization would not change degree distribution of complex networks but optimize it.

19. Figure 6 reflect the relationship between work efficiency and probability of removed nodes. Figure 6 work efficiency of scale-free networks

20. From figure 6 we knew that the work efficiency of topology optimization based on rank was better than BA model when suffered spitefully attack, but worse when suffered randomly attack.

21.  Conclusions  The research of scale-free networks, especially the frangibility,is very hot, and the results have largely changed and expanded our knowledge about anti-frangibility.  Scale-free networks is not only a common complex systems, it can be used as topology characteristics of the structure of complex systems model, and thus the research about anti- frangibility of scale-free networks is bound to promotion the research of complex systems, and become an importance research content in systems science.  Due to the structure, the scale-free networks were very robust when nodes were moved randomly but easily frangible when nodes were attacked spitefully.

22.  Conclusions (2)  The paper had analyzed the degree distribution, work efficiency and capability to endure disaster of complex networks and concluded that the exponent of degree distribution was 2.8, high work efficiency and better capability to endure spitefully disaster, the topology had been optimized.  In the past few years, although the study of scale-free have achieved significant progress, but most of the research results have failed to jump out of framework of Albert, namely research the anti-frangibility to different degree distribution under different attacks..  The actual network as the background, building models to anti-frangibility, analysis, optimization, control will be an important direction in the research of scale-free networks.

23. That’s all! Thank you very much! (Email: xukh@phy.ccnu.ecu.cn )