Scaling Outline Scaling Scaling—a Plenitude of Power Laws Scaling-at-large Scaling-at-large Scaling-at-large Principles of Complex Systems Allometry Allometry Definitions Definitions Course 300, Fall, 2008 Examples Examples Allometry History: Metabolism History: Metabolism Measuring exponents Measuring exponents Definitions History: River networks History: River networks Earlier theories Earlier theories Examples Geometric argument Geometric argument Prof. Peter Dodds Blood networks History: Metabolism Blood networks River networks River networks Conclusion Measuring exponents Conclusion Department of Mathematics & Statistics References References History: River networks University of Vermont Earlier theories Geometric argument Blood networks River networks Conclusion References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/114 Frame 2/114 Definitions Scaling Outline Scaling Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks All about scaling: History: River networks Earlier theories Earlier theories General observation: Geometric argument Geometric argument Blood networks Blood networks ◮ Definitions. River networks River networks Conclusion Conclusion Systems (complex or not) ◮ Examples. References References that cross many spatial and temporal scales ◮ How to measure your power-law relationship. often exhibit some form of scaling. ◮ Mechanisms giving rise to your power-laws. Frame 3/114 Frame 4/114
Definitions Scaling Definitions Scaling Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions Examples Examples A power law relates two variables x and y as follows: ◮ 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 y = cx α Earlier theories Earlier theories Geometric argument Geometric argument related as: Blood networks Blood networks River networks ℓ = cv 1 / 4 River networks Conclusion Conclusion References References ◮ Using [ · ] to indicate dimension, then ◮ α is the scaling exponent (or just exponent) ◮ ( α can be any number in principle but we will find [ c ] = [ l ] / [ V 1 / 4 ] = L / L 3 / 4 = L 1 / 4 . various restrictions.) ◮ c is the prefactor (which can be important!) Frame 5/114 Frame 6/114 Looking at data Scaling A beautiful, heart-warming example: Scaling Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions ◮ Power-law relationships are linear in log-log space: Examples α ≃ 1 . 23 Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents y = cx α History: River networks History: River networks gray Earlier theories Earlier theories Geometric argument Geometric argument matter: Blood networks Blood networks ⇒ log b y = α log b x + log b c River networks River networks ‘computing Conclusion Conclusion References References elements’ with slope equal to α , the scaling exponent. ◮ Much searching for straight lines on log-log or white double-logarithmic plots. matter: ◮ Good practice: Always, always, always use base 10. ‘wiring’ ◮ Talk only about orders of magnitude (powers of 10). from Zhang & Sejnowski, PNAS (2000) [26] Frame 7/114 Frame 8/114
Why is α ≃ 1 . 23? Scaling Why is α ≃ 1 . 23? Scaling Quantities (following Zhang and Sejnowski): Scaling-at-large Scaling-at-large Allometry Allometry ◮ G = Volume of gray matter (cortex/processors) Definitions Definitions Examples Examples ◮ W = Volume of white matter (wiring) History: Metabolism History: Metabolism Measuring exponents Measuring exponents ◮ T = Cortical thickness (wiring) History: River networks A rough understanding: History: River networks Earlier theories Earlier theories Geometric argument Geometric argument ◮ S = Cortical surface area Blood networks ◮ We are here: W ∝ G 4 / 3 / T Blood networks River networks River networks ◮ L = Average length of white matter fibers Conclusion Conclusion ◮ Observe weak scaling T ∝ G 0 . 10 ± 0 . 02 . References References ◮ p = density of axons on white matter/cortex interface ◮ (Implies S ∝ G 0 . 9 → convolutions fill space.) ◮ ⇒ W ∝ G 4 / 3 / T ∝ G 1 . 23 ± 0 . 02 A rough understanding: ◮ G ∼ ST (convolutions are okay) ◮ W ∼ 1 2 pSL ◮ G ∼ L 3 ← this is a little sketchy... ◮ Eliminate S and L to find W ∝ G 4 / 3 / T Frame 9/114 Frame 10/114 Why is α ≃ 1 . 23? Scaling Good scaling: Scaling Scaling-at-large Scaling-at-large General rules of thumb: Allometry Allometry Definitions Definitions Examples Examples ◮ High quality: scaling persists over History: Metabolism History: Metabolism Measuring exponents Measuring exponents three or more orders of magnitude History: River networks History: River networks Earlier theories Earlier theories for each variable. Geometric argument Geometric argument Blood networks Blood networks River networks River networks Conclusion Conclusion ◮ Medium quality: scaling persists over References References three or more orders of magnitude for only one variable and at least one for the other. Trickiness: ◮ Very dubious: scaling ‘persists’ over ◮ With V = G + W , some power laws must be less than an order of magnitude for both variables. approximations. ◮ Measuring exponents is a hairy business... Frame 11/114 Frame 12/114
Unconvincing scaling: Scaling Definitions Scaling Scaling-at-large Scaling-at-large Power laws are the signature of Allometry Allometry Average walking speed as a function of city Definitions Definitions scale invariance: Examples Examples population: History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks History: River networks Scale invariant ‘objects’ Earlier theories Earlier theories Geometric argument Geometric argument look the ‘same’ Two problems: Blood networks Blood networks River networks River networks when they are appropriately Conclusion Conclusion 1. use of natural log, and rescaled. References References 2. minute varation in dependent variable. ◮ Objects = geometric shapes, time series, functions, relationships, distributions,... ◮ ‘Same’ might be ‘statistically the same’ ◮ To rescale means to change the units of from Bettencourt et al. (2007) [3] ; otherwise very interesting! measurement for the relevant variables Frame 13/114 Frame 14/114 Scale invariance Scaling Scale invariance 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 ′ , Compare with y = ce − λ x : Measuring exponents Measuring exponents History: River networks History: River networks Earlier theories Earlier theories ◮ then Geometric argument Geometric argument ◮ If we rescale x as x = rx ′ , then Blood networks Blood networks r α y ′ = c ( rx ′ ) α River networks River networks Conclusion Conclusion y = ce − λ rx ′ References References ◮ ⇒ y ′ = cr α x ′ α r − α ◮ Original form cannot be recovered. ◮ ⇒ scale matters for the exponential. ◮ ⇒ y ′ = cx ′ α Frame 15/114 Frame 16/114
Scale invariance Scaling Definitions Scaling Scaling-at-large Scaling-at-large Allometry Allometry Definitions Definitions Examples Examples History: Metabolism History: Metabolism Measuring exponents Measuring exponents History: River networks History: River networks More on y = ce − λ x : Earlier theories Earlier theories Geometric argument Geometric argument Allometry ( ⊞ ): Blood networks Blood networks ◮ Say x 0 = 1 /λ is the characteristic scale. River networks River networks Conclusion Conclusion ◮ For x ≫ x 0 , y is small, [refers to] differential growth rates of the parts of a living References References while for x ≪ x 0 , y is large. organism’s body part or process. ◮ ⇒ More on this later with size distributions. Frame 17/114 Frame 19/114 Definitions: Scaling Definitions Scaling Isometry: Scaling-at-large Scaling-at-large dimensions scale Allometry Allometry Definitions Isometry versus Allometry: Definitions linearly with each Examples Examples History: Metabolism History: Metabolism other. Measuring exponents Measuring exponents ◮ Isometry = ‘same measure’ History: River networks History: River networks Earlier theories Earlier theories ◮ Allometry = ‘other measure’ Geometric argument Geometric argument Blood networks Blood networks River networks River networks Conclusion Conclusion References References Confusingly, we use allometric scaling to refer to both: 1. nonlinear scaling (e.g., x ∝ y 1 / 3 ) 2. and the relative scaling of different measures (e.g., resting heart rate as a function of body size) Allometry: dimensions scale nonlinearly. Frame 20/114 Frame 21/114
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