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Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, P(k), of primordial fluctuations as a function of wave number, k, is approximately constant, i.e. a flat spectrum.
The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
Estimating the power-law exponent of a scale-free network is typically done by using the maximum likelihood estimation with the degrees of a few uniformly sampled nodes. [3] However, since uniform sampling does not obtain enough samples from the important heavy-tail of the power law degree distribution, this method can yield a large bias and a ...
The study of power-laws is closely linked to the study of critical phenomena in physics, in which emergent properties and scale invariance are central and organizing concepts. These concepts resurface in the study of complex systems, [ 4 ] [ 5 ] and are of particular importance in the urban scaling framework.
Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is ...
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor.Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards ...
A change in scale is called a scale transformation. The renormalization group is intimately related to scale invariance and conformal invariance, symmetries in which a system appears the same at all scales (self-similarity). [a] As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system.
Bassingthwaighte's power law can be shown to directly relate to the variance-to-mean power law. Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution., [ 23 ] In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with gamma ...