Power laws and the AS-level internet topology

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

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Simulation of small-angle scattering curves by numerical Fourier transformation

Journal of Applied Crystallography ◽

10.1107/s002188980604550x ◽

2007 ◽

Vol 40 (1) ◽

pp. 16-25 ◽

Cited By ~ 37

Author(s):

Klaus Schmidt-Rohr

Keyword(s):

Form Factor ◽

Small Angle ◽

Fourier Transformation ◽

Three Dimensional ◽

Power Laws ◽

Numerical Approach ◽

Peak Position ◽

Two Dimensions ◽

Correlation Peak ◽

Scattering Intensity

A simple numerical approach for calculating theq-dependence of the scattering intensity in small-angle X-ray or neutron scattering (SAXS/SANS) is discussed. For a user-defined scattering density on a lattice, the scattering intensityI(q) (qis the modulus of the scattering vector) is calculated by three-dimensional (or two-dimensional) numerical Fourier transformation and spherical summation inqspace, with a simple smoothing algorithm. An exact and simple correction for continuous rather than discrete (lattice-point) scattering density is described. Applications to relatively densely packed particles in solids (e.g.nanocomposites) are shown, where correlation effects make single-particle (pure form-factor) calculations invalid. The algorithm can be applied to particles of any shape that can be defined on the chosen cubic lattice and with any size distribution, while those features pose difficulties to a traditional treatment in terms of form and structure factors. For particles of identical but potentially complex shapes, numerical calculation of the form factor is described. Long parallel rods and platelets of various cross-section shapes are particularly convenient to treat, since the calculation is reduced to two dimensions. The method is used to demonstrate that the scattering intensity from `randomly' parallel-packed long cylinders is not described by simple 1/qand 1/q4power laws, but at cylinder volume fractions of more than ∼25% includes a correlation peak. The simulations highlight that the traditional evaluation of the peak position overestimates the cylinder thickness by a factor of ∼1.5. It is also shown that a mix of various relatively densely packed long boards can produceI(q) ≃ 1/q, usually observed for rod-shaped particles, without a correlation peak.

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Revealing Novel Power Laws and Quantization in Electrospinning Considering Jet Splitting—Toward Predicting Fiber Diameter and Its Distribution Part II Experimental

Advanced Engineering Materials ◽

10.1002/adem.202001161 ◽

2021 ◽

pp. 2001161

Author(s):

Dirk W. Schubert ◽

Vincent Allen ◽

Ursula Dippel

Keyword(s):

Fiber Diameter ◽

Power Laws

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Epidemic growth and Griffiths effects on an emergent network of excited atoms

Nature Communications ◽

10.1038/s41467-020-20333-7 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

T. M. Wintermantel ◽

M. Buchhold ◽

S. Shevate ◽

M. Morgado ◽

Y. Wang ◽

...

Keyword(s):

Complex Systems ◽

Power Laws ◽

Excitation Density ◽

Griffiths Phase ◽

Many Body ◽

Excited Atoms ◽

Equilibrium Dynamics ◽

Many Body Systems ◽

Excitation Number ◽

Non Equilibrium

AbstractWhether it be physical, biological or social processes, complex systems exhibit dynamics that are exceedingly difficult to understand or predict from underlying principles. Here we report a striking correspondence between the excitation dynamics of a laser driven gas of Rydberg atoms and the spreading of diseases, which in turn opens up a controllable platform for studying non-equilibrium dynamics on complex networks. The competition between facilitated excitation and spontaneous decay results in sub-exponential growth of the excitation number, which is empirically observed in real epidemics. Based on this we develop a quantitative microscopic susceptible-infected-susceptible model which links the growth and final excitation density to the dynamics of an emergent heterogeneous network and rare active region effects associated to an extended Griffiths phase. This provides physical insights into the nature of non-equilibrium criticality in driven many-body systems and the mechanisms leading to non-universal power-laws in the dynamics of complex systems.

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