1. Emergence of double scaling law in complex systems 报告人：韩定定 华东师范大学 上海应用物理研究所

2. Although power law behaviors are pervasive in all kinds of studies, such a single property is usually insufficient to describe the whole distribution in real-world system, e.g. power law with exponential cut or double power behavior which we are interested here Word network Scientific collaborators network Chinese airline network People’s income distribution…

3. Some Reviews How to fit such distribution by a uniform function rather than treat two power seperately. (Non-extensive statistical theory and the combination of different power law functions) Geometric Brownian motion model coupled with exponential distributed evolution time, causing double-pareto-lognormal distribution. (income distribution W.J.Reed, Physica A 319, 469 (2002) ) the preferential attachment and the creation of new links between old nodes which increases linearly with evolution time, causing two different scaling exponents, -1.5 for upper tail and -3 for lower tail (word network S.N.Dorogovtsev, J.F.F.Mendes, arXiv:condmat/0105093v1 )

4. 1.In word network model the scaling exponents are fixed. 2.In Reed model,the relative increase rate of incomes is the same for everyone. 3.Have not been examined by the evolution of real-world network.

5. Main content Propose a simple model to generate double power-law based on fitness considerations. Generalized it to include noise fluctuations and explain its physical significance. Test it by an empirical study: Chinese airline network.

6. Our model and the results Two basic ingredients ： 1.Exponential growth of nodes 2. Normal distributed fitness Model:

7. Our model and the results

8. The key problem is how to prove the second part is power law For any ， namely the second part ， the integral of the left term from 0 to tc must be larger than that of the right term. The integral of the left term is exactly the degree distribution p(k) while the integral of the right termfollows asymptotically a power law, therefore distribution has a lower power law bound Then p(k) is expressed as where is valid for any when k converges to infinite ， therefore we have ： And then

9. The second power exponent Note that when it naturally degenerates to be a cutoff

10. Generalized model Fluctuation consideration Modified equation: where dw is white noise and is the standard variance of fluctuations. is a parameter that represents the relative contribution of the noise and the fitness. Noise fluctuations do not change the distribution qualitatively but contribute to the second power exponent.

11. Physical significance Evolution = leading ingredients (fitness term) + combination of other minor ingredients (fluctuations) if the evolution is totally governed by the leading ingredient, it performs a deterministic picture (order). If there is no apparent leading ingredients, then a random process (disorder)

12. An example: Chinese airline network 1.Leading ingredient: economy (GDP) 2.We investigate the evolution by looking at the correlation between the GDP of a node and its corresponding degree 3. The evolution can also be investigated without introducing any hidden variable, but this method can neither help to distinguish the identity of the fitness nor provide useful information of the corresponding parameter.

13. Some results

15. About D(t) What we are interested in is the increment of

16. Exact the form of our model The identity of the fitness is GDP growth rate Comparable to the empirical value 0.51 Good agreement with the second power 2.7