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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
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…
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)
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.
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.
Our model and the results
Two basic ingredients：
1.Exponential growth of nodes
2. Normal distributed fitness
Model:
Our model and the results
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
for any
And then
where
when k converges to infinite，therefore we have：
is valid
The second power exponent
Note that when
it naturally degenerates to
be a cutoff
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.
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)
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.
Some results
Some results
About D(t)
What we are interested in is the increment of
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