Scaling Scaling Outline Scaling—a Plenitude of Power Laws Scaling-at-large Scaling-at-large Principles of Complex Systems Allometry Allometry CSYS/MATH 300, Fall, 2010 Definitions Definitions Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks All about scaling: History: River networks Prof. Peter Dodds Earlier theories Earlier theories Geometric argument Geometric argument Blood networks ◮ Definitions. Blood networks River networks River networks Department of Mathematics & Statistics Conclusion Conclusion Center for Complex Systems ◮ Examples. References References Vermont Advanced Computing Center ◮ How to measure your power-law relationship. University of Vermont ◮ Mechanisms giving rise to your power-laws. Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . 1 of 126 4 of 126 Scaling Scaling Outline Definitions Scaling-at-large Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions Examples Examples Allometry A power law relates two variables x and y as follows: History: Metabolism History: Metabolism Definitions Measuring exponents Measuring exponents History: River networks History: River networks y = cx α Examples Earlier theories Earlier theories Geometric argument Geometric argument Blood networks Blood networks History: Metabolism River networks River networks Measuring exponents Conclusion Conclusion References References History: River networks ◮ α is the scaling exponent (or just exponent) Earlier theories ◮ ( α can be any number in principle but we will find Geometric argument various restrictions.) Blood networks ◮ c is the prefactor (which can be important!) River networks Conclusion References 2 of 126 5 of 126 Definitions Scaling Definitions Scaling Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions Examples Examples ◮ The prefactor c must balance dimensions. History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks History: River networks ◮ eg., length ℓ and volume v of common nails are Earlier theories Earlier theories General observation: Geometric argument Geometric argument related as: Blood networks Blood networks River networks ℓ = cv 1 / 4 River networks Conclusion Conclusion Systems (complex or not) References References ◮ Using [ · ] to indicate dimension, then that cross many spatial and temporal scales often exhibit some form of scaling. [ c ] = [ l ] / [ V 1 / 4 ] = L / L 3 / 4 = L 1 / 4 . 3 of 126 6 of 126
Scaling Scaling Looking at data Why is α ≃ 1 . 23? Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions ◮ Power-law relationships are linear in log-log space: Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents y = cx α History: River networks A rough understanding: History: River networks Earlier theories Earlier theories Geometric argument Geometric argument Blood networks ◮ We are here: W ∝ G 4 / 3 / T Blood networks ⇒ log b y = α log b x + log b c River networks River networks Conclusion Conclusion ◮ Observe weak scaling T ∝ G 0 . 10 ± 0 . 02 . References References with slope equal to α , the scaling exponent. ◮ (Implies S ∝ G 0 . 9 → convolutions fill space.) ◮ Much searching for straight lines on log-log or ◮ ⇒ W ∝ G 4 / 3 / T ∝ G 1 . 23 ± 0 . 02 double-logarithmic plots. ◮ Good practice: Always, always, always use base 10. ◮ Talk only about orders of magnitude (powers of 10). 7 of 126 10 of 126 Scaling Why is α ≃ 1 . 23? Scaling A beautiful, heart-warming example: Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions α ≃ 1 . 23 Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks History: River networks gray Earlier theories Earlier theories Geometric argument Geometric argument matter: Blood networks Blood networks River networks River networks ‘computing Conclusion Conclusion elements’ References References white matter: ‘wiring’ Trickiness: ◮ With V = G + W , some power laws must be approximations. from Zhang & Sejnowski, PNAS (2000) [41] ◮ Measuring exponents is a hairy business... 8 of 126 11 of 126 Why is α ≃ 1 . 23? Scaling Good scaling: Scaling Quantities (following Zhang and Sejnowski): Scaling-at-large Scaling-at-large General rules of thumb: Allometry Allometry ◮ G = Volume of gray matter (cortex/processors) Definitions Definitions Examples Examples ◮ High quality: scaling persists over ◮ W = Volume of white matter (wiring) History: Metabolism History: Metabolism Measuring exponents Measuring exponents three or more orders of magnitude ◮ T = Cortical thickness (wiring) History: River networks History: River networks Earlier theories Earlier theories for each variable. Geometric argument Geometric argument ◮ S = Cortical surface area Blood networks Blood networks River networks River networks ◮ L = Average length of white matter fibers Conclusion Conclusion ◮ Medium quality: scaling persists over References References ◮ p = density of axons on white matter/cortex interface three or more orders of magnitude for only one variable and at least one for the other. A rough understanding: ◮ G ∼ ST (convolutions are okay) ◮ Very dubious: scaling ‘persists’ over less than an order of magnitude ◮ W ∼ 1 2 pSL for both variables. ◮ G ∼ L 3 ← this is a little sketchy... ◮ Eliminate S and L to find W ∝ G 4 / 3 / T 9 of 126 12 of 126
Scaling Scaling Unconvincing scaling: Scale invariance Scaling-at-large Scaling-at-large Allometry Allometry Average walking speed as a function of city Definitions Definitions Examples Examples population: History: Metabolism History: Metabolism Compare with y = ce − λ x : Measuring exponents Measuring exponents History: River networks History: River networks Earlier theories Earlier theories Geometric argument Geometric argument ◮ If we rescale x as x = rx ′ , then Two problems: Blood networks Blood networks River networks River networks Conclusion Conclusion 1. use of natural log, and y = ce − λ rx ′ References References 2. minute varation in ◮ Original form cannot be recovered. dependent variable. ◮ ⇒ scale matters for the exponential. from Bettencourt et al. (2007) [4] ; otherwise very interesting! 13 of 126 16 of 126 Scaling Scaling Definitions Scale invariance Scaling-at-large Scaling-at-large Power laws are the signature of Allometry Allometry Definitions Definitions scale invariance: Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks More on y = ce − λ x : History: River networks Scale invariant ‘objects’ Earlier theories Earlier theories Geometric argument Geometric argument look the ‘same’ Blood networks Blood networks ◮ Say x 0 = 1 /λ is the characteristic scale. River networks River networks when they are appropriately Conclusion Conclusion ◮ For x ≫ x 0 , y is small, rescaled. References References while for x ≪ x 0 , y is large. ◮ Objects = geometric shapes, time series, functions, ◮ ⇒ More on this later with size distributions. relationships, distributions,... ◮ ‘Same’ might be ‘statistically the same’ ◮ To rescale means to change the units of measurement for the relevant variables 14 of 126 17 of 126 Scale invariance Scaling Definitions Scaling Scaling-at-large Scaling-at-large Allometry Allometry Our friend y = cx α : Definitions Definitions Examples Examples History: Metabolism History: Metabolism ◮ If we rescale x as x = rx ′ and y as y = r α y ′ , Measuring exponents Measuring exponents Allometry: ( ⊞ ) History: River networks History: River networks Earlier theories Earlier theories ◮ then Geometric argument Geometric argument Blood networks Blood networks r α y ′ = c ( rx ′ ) α River networks River networks refers to differential growth rates of the parts of a living Conclusion Conclusion References References organism’s body part or process. ◮ ⇒ y ′ = cr α x ′ α r − α ◮ First proposed by Huxley and Teissier, Nature, 1936 “Terminology of relative growth” [21] ◮ ⇒ y ′ = cx ′ α 15 of 126 19 of 126
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