Scaling A wonderful treatise on scaling: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion McMahon and References Bonner, 1983 [18] Frame 23/117
Scaling For the following slide: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 24/117 p. 2, McMahon and Bonner [18]
Scaling The many scales of life: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 25/117 p. 2, McMahon and Bonner [18]
Scaling For the following slide: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 26/117 p. 2, McMahon and Bonner [18]
Scaling The many scales of life: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 27/117 p. 3, McMahon and Bonner [18]
Scaling For the following slide: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 28/117 p. 2, McMahon and Bonner [18]
Scaling The many scales of life: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 29/117 p. 3, McMahon and Bonner [18]
Scaling Size range and cell differentiation: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 30/117 [18]
Scaling Non-uniform growth: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 31/117 p. 32, McMahon and Bonner [18]
Scaling Non-uniform growth—arm length versus height: Scaling-at-large Allometry Definitions Good example of a break in scaling: Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References A crossover in scaling occurs around a height of 1 metre. p. 32, McMahon and Bonner [18] Frame 32/117
Weightlifting: M worldrecord ∝ M 2 / 3 Scaling lifter Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Idea: Power ∼ cross-sectional area of isometric lifters. p. 53, McMahon and Bonner [18] Frame 33/117
Titanothere horns: L horn ∼ L 4 Scaling skull Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 34/117 p. 36, McMahon and Bonner [18]
Scaling The allometry of nails: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References ◮ Diameter ∝ Mass 3 / 8 ◮ Length ∝ Mass 1 / 4 ◮ Diameter ∝ Length 2 / 3 p. 58–59, McMahon and Bonner [18] Frame 35/117
Scaling The allometry of nails: Scaling-at-large Allometry Definitions Examples History: Metabolism A buckling instability?: Measuring exponents History: River networks Earlier theories ◮ Physics/Engineering result: Columns buckle under a Geometric argument Blood networks load which depends on d 4 /ℓ 2 . River networks Conclusion ◮ To drive nails in, resistive force ∝ nail circumference References = π d . ◮ Match forces independent of nail size: d 4 /ℓ 2 ∝ d . ◮ Leads to d ∝ ℓ 2 / 3 . Frame 36/117
Rowing: Speed ∝ (number of rowers) 1 / 9 Scaling Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 37/117
Scaling Scaling in Cities: Scaling-at-large Allometry Definitions Examples ◮ “Growth, innovation, scaling, and the pace of life in History: Metabolism Measuring exponents cities” History: River networks Earlier theories Bettencourt et al., PNAS, 2007. [3] Geometric argument Blood networks ◮ Quantified levels of River networks Conclusion ◮ Infrastructure References ◮ Wealth ◮ Crime levels ◮ Disease ◮ Energy consumption as a function of city size N (population). Frame 38/117
Scaling Scaling in Cities: Table 1. Scaling exponents for urban indicators vs. city size Scaling-at-large Y � 95% CI Adj- R 2 Observations Country–year Allometry New patents 1.27 � 1.25,1.29 � 0.72 331 U.S. 2001 Definitions Inventors 1.25 � 1.22,1.27 � 0.76 331 U.S. 2001 Examples Private R&D employment 1.34 � 1.29,1.39 � 0.92 266 U.S. 2002 History: Metabolism � Supercreative � employment 1.15 � 1.11,1.18 � 0.89 287 U.S. 2003 Measuring exponents R&D establishments 1.19 � 1.14,1.22 � 0.77 287 U.S. 1997 History: River networks Earlier theories R&D employment 1.26 � 1.18,1.43 � 0.93 295 China 2002 Geometric argument Total wages 1.12 � 1.09,1.13 � 0.96 361 U.S. 2002 Blood networks Total bank deposits 1.08 � 1.03,1.11 � 0.91 267 U.S. 1996 River networks GDP 1.15 � 1.06,1.23 � 0.96 295 China 2002 Conclusion GDP 1.26 � 1.09,1.46 � 0.64 196 EU 1999–2003 GDP 1.13 � 1.03,1.23 � 0.94 37 Germany 2003 References Total electrical consumption 1.07 � 1.03,1.11 � 0.88 392 Germany 2002 New AIDS cases 1.23 � 1.18,1.29 � 0.76 93 U.S. 2002–2003 Serious crimes 1.16 [1.11, 1.18] 0.89 287 U.S. 2003 Total housing 1.00 � 0.99,1.01 � 0.99 316 U.S. 1990 Total employment 1.01 � 0.99,1.02 � 0.98 331 U.S. 2001 Household electrical consumption 1.00 � 0.94,1.06 � 0.88 377 Germany 2002 Household electrical consumption 1.05 � 0.89,1.22 � 0.91 295 China 2002 Household water consumption 1.01 � 0.89,1.11 � 0.96 295 China 2002 Gasoline stations 0.77 � 0.74,0.81 � 0.93 318 U.S. 2001 Gasoline sales 0.79 � 0.73,0.80 � 0.94 318 U.S. 2001 Length of electrical cables 0.87 � 0.82,0.92 � 0.75 380 Germany 2002 Road surface 0.83 � 0.74,0.92 � 0.87 29 Germany 2002 Data sources are shown in SI Text . CI, confidence interval; Adj- R 2 , adjusted R 2 ; GDP, gross domestic product. Frame 39/117
Scaling Scaling in Cities: Scaling-at-large Intriguing findings: Allometry ◮ Global supply costs scale sublinearly with N ( β < 1). Definitions Examples History: Metabolism ◮ Returns to scale for infrastructure. Measuring exponents History: River networks ◮ Total individual costs scale linearly with N ( β = 1) Earlier theories Geometric argument ◮ Individuals consume similar amounts independent of Blood networks River networks city size. Conclusion ◮ Social quantities scale superlinearly with N ( β > 1) References ◮ Creativity (# patents), wealth, disease, crime, ... Density doesn’t seem to matter... ◮ Surprising given that across the world, we observe two orders of magnitude variation in area covered by agglomerations of fixed populations. Frame 40/117
Ecology—Species-area law: N species ∝ A β Scaling Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Allegedly (data is messy): Geometric argument Blood networks River networks ◮ On islands: β ≈ 1 / 4. Conclusion References ◮ On continuous land: β ≈ 1 / 8. Frame 41/117
Scaling A focus: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks ◮ How much energy do organisms need to live? River networks Conclusion ◮ And how does this scale with organismal size? References Frame 42/117
Scaling Animal power Scaling-at-large Allometry Fundamental biological and ecological constraint: Definitions Examples History: Metabolism Measuring exponents P = c M α History: River networks Earlier theories Geometric argument Blood networks P = basal metabolic rate River networks Conclusion M = organismal body mass References Frame 44/117
P = c M α Scaling Scaling-at-large Prefactor c depends on body plan and body temperature: Allometry Definitions Examples 39–41 ◦ C Birds History: Metabolism Measuring exponents 36–38 ◦ C Eutherian Mammals History: River networks Earlier theories 34–36 ◦ C Marsupials Geometric argument Blood networks 30–31 ◦ C Monotremes River networks Conclusion References Frame 45/117
Scaling What one might expect: Scaling-at-large α = 2 / 3 because . . . Allometry Definitions Examples ◮ Dimensional analysis suggests History: Metabolism Measuring exponents History: River networks an energy balance surface law: Earlier theories Geometric argument Blood networks P ∝ S ∝ V 2 / 3 ∝ M 2 / 3 River networks Conclusion References ◮ Lognormal fluctuations: Gaussian fluctuations in log P around log cM α . ◮ Stefan-Boltzmann relation for radiated energy: d E d t = σε ST 4 Frame 46/117
Scaling The prevailing belief of the church of quarterology Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument α = 3 / 4 Blood networks River networks Conclusion References P ∝ M 3 / 4 Huh? Frame 47/117
Scaling Related putative scalings: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks ◮ number of capillaries ∝ M 3 / 4 Earlier theories Geometric argument ◮ time to reproductive maturity ∝ M 1 / 4 Blood networks River networks Conclusion ◮ heart rate ∝ M − 1 / 4 References ◮ cross-sectional area of aorta ∝ M 3 / 4 ◮ population density ∝ M − 3 / 4 Frame 48/117
Scaling The great ‘law’ of heartbeats: Scaling-at-large Assuming: Allometry ◮ Average lifespan ∝ M β Definitions Examples History: Metabolism ◮ Average heart rate ∝ M − β Measuring exponents History: River networks ◮ Irrelevant but perhaps β = 1 / 4. Earlier theories Geometric argument Blood networks River networks Conclusion Then: References ◮ Average number of heart beats in a lifespan ≃ (Average lifespan) × (Average heart rate) ∝ M β − β ∝ M 0 ◮ Number of heartbeats per life time is independent of organism size! ◮ ≈ 1.5 billion.... Frame 49/117
Scaling History Scaling-at-large Allometry Definitions Examples 1840’s: Sarrus and Rameaux [22] first suggested α = 2 / 3. History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 50/117
Scaling History Scaling-at-large Allometry Definitions Examples 1883: Rubner [21] found α ≃ 2 / 3. History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 51/117
Scaling History Scaling-at-large Allometry Definitions Examples 1930’s: Brody, Benedict study mammals. [6] History: Metabolism Measuring exponents Found α ≃ 0 . 73 (standard). History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 52/117
Scaling History Scaling-at-large Allometry Definitions Examples 1932: Kleiber analyzed 13 mammals. [15] History: Metabolism Measuring exponents Found α = 0 . 76 and suggested α = 3 / 4. History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 53/117
Scaling History Scaling-at-large Allometry Definitions 1950/1960: Hemmingsen [12, 13] Examples History: Metabolism Extension to unicellular organisms. Measuring exponents History: River networks α = 3 / 4 assumed true. Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 54/117
Scaling History Scaling-at-large Allometry Definitions 1964: Troon, Scotland: [4] Examples History: Metabolism 3rd symposium on energy metabolism. Measuring exponents History: River networks α = 3 / 4 made official . . . . . . 29 to zip. Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 55/117
Scaling Today Scaling-at-large Allometry ◮ 3/4 is held by many to be the one true exponent. Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks In the Beat of a Heart: Life, Energy, and River networks Conclusion the Unity of Nature —by John Whitfield References ◮ But—much controversy... ◮ See ‘Re-examination of the “3/4-law” of metabolism’ Dodds, Rothman, and Weitz [9] Frame 56/117
Scaling Some data on metabolic rates Scaling-at-large 3.5 Allometry Definitions B = 0.026 M 0.668 3 Examples History: Metabolism 2.5 Measuring exponents ◮ Heusner’s History: River networks 2 Earlier theories data Geometric argument 1.5 [source=/home/dodds/work/biology/allometry/heusner/figures/figheusner391.ps] Blood networks log 10 B (1991) [14] River networks 1 Conclusion ◮ 391 Mammals 0.5 References 0 ◮ blue line: 2/3 −0.5 ◮ red line: 3/4. −1 ◮ ( B = P ) −1.5 [10−Dec−2001 peter dodds] 0 1 2 3 4 5 6 7 log 10 M Frame 57/117
Scaling Some data on metabolic rates 2 Scaling-at-large B = 0.041 M 0.664 Allometry 1.5 Definitions Examples ◮ Bennett and 1 History: Metabolism Measuring exponents Harvey’s data History: River networks log 10 B 0.5 Earlier theories (1987) [2] Geometric argument Blood networks ◮ 398 birds River networks 0 Conclusion ◮ blue line: 2/3 References −0.5 ◮ red line: 3/4. −1 ◮ ( B = P ) −1.5 0 1 2 3 4 5 log 10 M Passerine vs. non-passerine... Frame 58/117
Scaling Linear regression Scaling-at-large Allometry Definitions Examples Important: History: Metabolism Measuring exponents History: River networks ◮ Ordinary Least Squares (OLS) Linear regression is Earlier theories Geometric argument Blood networks only appropriate for analyzing a dataset { ( x i , y i ) } River networks when we know the x i are measured without error. Conclusion References ◮ Here we assume that measurements of mass M have less error than measurements of metabolic rate B . ◮ Linear regression assumes Gaussian errors. Frame 60/117
Scaling Measuring exponents Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents More on regression: History: River networks Earlier theories If (a) we don’t know what the errors of either variable are, Geometric argument Blood networks River networks or (b) no variable can be considered independent, Conclusion References then we need to use Standardized Major Axis Linear Regression. (aka Reduced Major Axis = RMA.) Frame 61/117
Scaling Measuring exponents Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories For Standardized Major Axis Linear Regression: Geometric argument Blood networks River networks slope SMA = standard deviation of y data Conclusion References standard deviation of x data Very simple! Frame 62/117
Scaling Measuring exponents Scaling-at-large Allometry Definitions Relationship to ordinary least squares regression is Examples History: Metabolism simple: Measuring exponents History: River networks Earlier theories r − 1 × slope OLS y on x slope SMA = Geometric argument Blood networks River networks = r × slope OLS x on y Conclusion References where r = standard correlation coefficient: � n i = 1 ( x i − ¯ x )( y i − ¯ y ) r = �� n �� n i = 1 ( x i − ¯ x ) 2 i = 1 ( y i − ¯ y ) 2 Frame 63/117
Scaling Heusner’s data, 1991 (391 Mammals) Scaling-at-large range of M N α ˆ Allometry Definitions Examples ≤ 0 . 1 kg 0 . 678 ± 0 . 038 167 History: Metabolism Measuring exponents History: River networks Earlier theories ≤ 1 kg 276 0 . 662 ± 0 . 032 Geometric argument Blood networks River networks Conclusion ≤ 10 kg 357 0 . 668 ± 0 . 019 References ≤ 25 kg 366 0 . 669 ± 0 . 018 ≤ 35 kg 371 0 . 675 ± 0 . 018 ≤ 350 kg 389 0 . 706 ± 0 . 016 ≤ 3670 kg 391 0 . 710 ± 0 . 021 Frame 64/117
Scaling Bennett and Harvey, 1987 (398 birds) M max N α ˆ Scaling-at-large Allometry ≤ 0 . 032 162 0 . 636 ± 0 . 103 Definitions Examples History: Metabolism Measuring exponents ≤ 0 . 1 236 0 . 602 ± 0 . 060 History: River networks Earlier theories Geometric argument Blood networks ≤ 0 . 32 290 0 . 607 ± 0 . 039 River networks Conclusion References ≤ 1 334 0 . 652 ± 0 . 030 ≤ 3 . 2 371 0 . 655 ± 0 . 023 ≤ 10 391 0 . 664 ± 0 . 020 ≤ 32 0 . 665 ± 0 . 019 396 Frame 65/117 ≤ 100 398 0 . 664 ± 0 . 019
Scaling Hypothesis testing Scaling-at-large Test to see if α ′ is consistent with our data { ( M i , B i ) } : Allometry Definitions Examples H 0 : α = α ′ and H 1 : α � = α ′ . History: Metabolism Measuring exponents History: River networks Earlier theories ◮ Assume each B i (now a random variable) is normally Geometric argument Blood networks distributed about α ′ log 10 M i + log 10 c . River networks Conclusion ◮ Follows that the measured α for one realization References obeys a t distribution with N − 2 degrees of freedom. ◮ Calculate a p -value: probability that the measured α is as least as different to our hypothesized α ′ as we observe. ◮ (see, for example, DeGroot and Scherish, “Probability and Statistics” [7] ) Frame 66/117
Scaling Revisiting the past—mammals Scaling-at-large Allometry Definitions Full mass range: Examples History: Metabolism N α ˆ p 2 / 3 p 3 / 4 Measuring exponents History: River networks Earlier theories < 10 − 6 Geometric argument Kleiber 13 0.738 0.11 Blood networks River networks Conclusion < 10 − 4 < 10 − 2 Brody 35 0.718 References < 10 − 6 < 10 − 5 Heusner 391 0.710 < 10 − 15 Bennett 398 0.664 0.69 and Harvey Frame 67/117
Scaling Revisiting the past—mammals Scaling-at-large M ≤ 10 kg: Allometry N α ˆ p 2 / 3 p 3 / 4 Definitions Examples History: Metabolism Kleiber 5 0.667 0.99 0.088 Measuring exponents History: River networks Earlier theories < 10 − 3 < 10 − 3 Brody 26 0.709 Geometric argument Blood networks River networks Conclusion < 10 − 15 Heusner 357 0.668 0.91 References M ≥ 10 kg: N α ˆ p 2 / 3 p 3 / 4 < 10 − 4 Kleiber 8 0.754 0.66 < 10 − 3 Brody 9 0.760 0.56 < 10 − 12 < 10 − 7 Heusner 34 0.877 Frame 68/117
Scaling Fluctuations—Kolmogorov-Smirnov test Scaling-at-large [07−Nov−1999 peter dodds] 4 Allometry Definitions 20 bins 3.5 Examples History: Metabolism Measuring exponents 3 History: River networks P( log 10 B / M 2/3 ) Earlier theories Geometric argument [source=/home/dodds/work/biology/allometry/heusner/figures/figmetascalingfn2.ps] 2.5 Blood networks River networks Conclusion 2 References 1.5 1 0.5 0 −0.5 0 0.5 log 10 B / M 2/3 P ( B | M ) = 1 / M 2 / 3 f ( B / M 2 / 3 ) Frame 69/117
Scaling Analysis of residuals Scaling-at-large Allometry Definitions Examples 1. Presume an exponent of your choice: 2/3 or 3/4. History: Metabolism Measuring exponents 2. Fit the prefactor (log 10 c ) and then examine the History: River networks Earlier theories residuals: Geometric argument Blood networks River networks r i = log 10 B i − ( α ′ log 10 M i − log 10 c ) . Conclusion References 3. H 0 : residuals are uncorrelated H 1 : residuals are correlated. 4. Measure the correlations in the residuals and compute a p -value. Frame 70/117
Scaling Analysis of residuals Scaling-at-large We use the spiffing Spearman Rank-Order Correlation Allometry Definitions Cofficient. Examples History: Metabolism Measuring exponents Basic idea: History: River networks Earlier theories Geometric argument ◮ Given { ( x i , y i ) } , rank the { x i } and { y i } separately Blood networks River networks from smallest to largest. Call these ranks R i and S i . Conclusion ◮ Now calculate correlation coefficient for ranks, r s : References ◮ R )( S i − ¯ � n i = 1 ( R i − ¯ S ) r s = �� n �� n i = 1 ( R i − ¯ i = 1 ( S i − ¯ R ) 2 S ) 2 ◮ Perfect correlation: x i ’s and y i ’s both increase monotonically. Frame 71/117
Scaling Analysis of residuals Scaling-at-large Allometry Definitions We assume all rank orderings are equally likely: Examples History: Metabolism Measuring exponents ◮ r s is distributed according to a Student’s distribution History: River networks Earlier theories with N − 2 degrees of freedom. Geometric argument Blood networks River networks ◮ Excellent feature: Non-parametric—real distribution Conclusion of x ’s and y ’s doesn’t matter. References ◮ Bonus: works for non-linear monotonic relationships as well. ◮ See “Numerical Recipes in C/Fortran” which contains many good things. [20] Frame 72/117
Scaling Analysis of residuals—mammals Scaling-at-large Allometry Definitions 0 0 Examples (a) (b) History: Metabolism −1 −1 Measuring exponents History: River networks −2 −2 Earlier theories Geometric argument −3 −3 Blood networks River networks −4 −4 log 10 p Conclusion 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 References 0 0 (c) (d) −1 −1 −2 −2 −3 −3 −4 −4 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 α ’ (a) M < 3 . 2 kg, (b) M < 10 kg, (c) M < 32 kg, (d) all mammals. Frame 73/117
Scaling Analysis of residuals—birds Scaling-at-large Allometry Definitions 0 0 Examples (a) (b) History: Metabolism −1 −1 Measuring exponents History: River networks −2 −2 Earlier theories Geometric argument −3 −3 Blood networks River networks −4 −4 log 10 p Conclusion 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 References 0 0 (c) (d) −1 −1 −2 −2 −3 −3 −4 −4 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 α ’ (a) M < . 1 kg, (b) M < 1 kg, (c) M < 10 kg, (d) all birds. Frame 74/117
0 L l ? 0 l 0 a Scaling Basic basin quantities: a , l , L � , L ⊥ : 0 L a k Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents ◮ a = drainage History: River networks L = L k Earlier theories basin area Geometric argument Blood networks River networks ◮ ℓ = length of Conclusion L ? longest (main) References stream ◮ L = L � = longitudinal length of basin Frame 76/117
Scaling River networks Scaling-at-large Allometry Definitions ◮ 1957: J. T. Hack [11] Examples History: Metabolism “Studies of Longitudinal Stream Profiles in Virginia Measuring exponents History: River networks and Maryland” Earlier theories Geometric argument ℓ ∼ a h Blood networks River networks Conclusion h ∼ 0 . 6 References ◮ Anomalous scaling: we would expect h = 1 / 2... ◮ Subsequent studies: 0 . 5 � h � 0 . 6 ◮ Another quest to find universality/god... ◮ A catch: studies done on small scales. Frame 77/117
Fig. 1. Without a scale bar it is almost impossible to de- termine even the approxi- mate scale of a topographic map. The upper two maps show adjacent drainage ba- sins in the Oregon Coast Fig. 3. The coherence of the data in Fig. 2 across Range and illustrate the ef- 11 orders of magnitude indicates a geometric fect of depicting an area of similarity between small drainage basins and the similar topography at diffier- larger drainage basins that contain them. Al- ent scales. The map on the though the variance about the trend in Fig. 2 area four right covers an indicates a range in individual basin shapes, this times as large as, and has general relation apparently characterizes the land- twice the contour interval scape down to the finest scale of convergent of, the map on the left. The topography. lower two maps depict very different landscapes, and de- tailed mapping was done to - -- resolve the finest scale val- that of the topographically divergent ridges - leys, which determine the that separate these fine-scale valleys. extent, or scale, of landscape Equation 1 differs, however, from the dissection. The map on the left shows a portion of a/ relation between the mainstream length small badlands area at Perth and drainage area first reported by Hack Amboy, New Jersey (28) (11), in which basin area increases as L`. (scale bar represents 2 m; Many subsequent workers interpreted sim- contour interval is 0.3 in). ilar relations as indicating that drainage The map on the right shows a portion of the San Gabriel network planform geometry changes with Mountains of southern Cal- increasing scale. Relations between main- ifornia (20) (scale bar repre- stream length and drainage area also have sents 100 m; contour inter- been used to infer the fractal dimension of val is 15 in). Dashed fines on individual channels and channel networks both lower maps represent the limit of original map- 16). Mueller (15), however, reported (1, ping. The drainage basin outlet on each map is oriented toward the bottom of the page. All four maps that the exponent in the relation of main- suggest a limit to landscape dissection, defined by the size of the hilislopes, separating valleys. This stream length to drainage area is not con- apparent limit, however, only corresponds to the extent of valley dissection definable in the field for the stant, but decreases from 0.6 to -0.5 with case of the lower two maps. increasing network size, and Hack (11) noted that the exponent in this relation varies for individual drainage networks. where L and A are expressed in meters. This We collected data from small drainage relation is well approximated by the simple, We cannot compare our data more quanti- basins in a variety of geologic settings that tatively with those reported by others be- isometric relation represent a range in climate and vegetation cause the mainstream length will diverge (4, 5). We measured the drainage area (A), (3 A)0 5 (2) L from the basin length in proportion to the basin length (L), and local slope (S) for area upslope of the stream head. We sus- Inclusion of reported drainage area and locations in convergent topography along pect that the difference in the relations mainstream length data from larger net- at channel low-order channel networks, derived from our data and those reported works (11-15) provides a composite data heads, and along unchanneled valleys in set that also is reasonably fit (5) by this previously reflects variation in the head- drainage basins where we had mapped the ward extent of the stream network depicted relation. The data span a range of more channel networks in the field (4, 5). Drain- on maps of varying scale (17) as well as than 11 orders of magnitude in basin area, age area was defined as the area upslope of downstream variations in both channel sin- from unchanneled hillside depressions to the measurement location, basin length was uosity (14) and drainage density (18). The the world's largest rivers (Fig. 2). This defined as the length along the main valley general scale independence indicated in axis to the drainage divide, and local slope relation suggests that there is a basic geo- Fig. 2 suggests that landscape dissection metric similarity between drainage basins was measured in the field. The structural results in an integrated network of valleys and the smaller basins contained within relation of drainage area to basin length (10) that capture geometrically similar drainage them that holds down to the finest scale to for our composite data set is basins at scales ranging from the largest which the landscape is dissected (Fig. 3). Scaling Large-scale networks rivers to the finest scale valleys. Within this In the field this scale is easily recognized as L = 1.78 A49 (1) scale range there appears to be little inher- (1992) Montgomery and Dietrich [19] : Scaling-at-large ent to the channel network and to the Allometry corresponding shape of the drainage area it Definitions captures that provides reference to an ab- Examples History: Metabolism solute scale. E Measuring exponents Fig. 2. Basin length versus drainage History: River networks Nonetheless, field studies in semiarid to area for unchanneled valleys, source Earlier theories areas, and low-order channels mapped Geometric argument humid regions demonstrate that there is a 5 Blood networks this study (0) and mainstream finite extent to the branching channel net- in River networks U Conclusion length versus drainage area data report- work (4, 5, 19-22). Channels do not occupy c ed for large channel networks (0). References the entire landscape; rather, they typically Sources of mainstream length data are begin at the foot of an unchanneled valley, given in (5). Drainage area (m2) REPORT 827 ◮ Composite data set: includes everything from 14 FEBRUARY 1992 unchanneled valleys up to world’s largest rivers. ◮ Estimated fit: L ≃ 1 . 78 a 0 . 49 ◮ Mixture of basin and main stream lengths. Frame 78/117
Scaling World’s largest rivers only: Scaling-at-large Allometry 4 10 Definitions (mi) Examples History: Metabolism Measuring exponents l length History: River networks Earlier theories 3 Geometric argument 10 stream Blood networks River networks area a (sq mi) Conclusion main References 2 10 4 5 6 7 10 10 10 10 ◮ Data from Leopold (1994) [16, 8] ◮ Estimate of Hack exponent: h = 0 . 50 ± 0 . 06 Frame 79/117
Scaling Earlier theories Scaling-at-large Allometry Definitions Examples Building on the surface area idea... History: Metabolism Measuring exponents ◮ Blum (1977) [5] speculates on four-dimensional History: River networks Earlier theories Geometric argument biology: Blood networks River networks P ∝ M ( d − 1 ) / d Conclusion References ◮ d = 3 gives α = 2 / 3 ◮ d = 4 gives α = 3 / 4 ◮ So we need another dimension... ◮ Obviously, a bit silly. Frame 81/117
Scaling Earlier theories Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents Building on the surface area idea: History: River networks Earlier theories Geometric argument ◮ McMahon (70’s, 80’s): Elastic Similarity [17, 18] Blood networks River networks ◮ Idea is that organismal shapes scale allometrically Conclusion References with 1/4 powers (like nails and trees...) ◮ Appears to be true for ungulate legs. ◮ Metabolism and shape never properly connected. Frame 82/117
Scaling Nutrient delivering networks: ◮ 1960’s: Rashevsky considers blood networks and Scaling-at-large finds a 2 / 3 scaling. Allometry ◮ 1997: West et al. [25] use a network story to find 3 / 4 Definitions Examples scaling. History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Frame 83/117
Scaling Nutrient delivering networks: Scaling-at-large Allometry West et al.’s assumptions: Definitions Examples History: Metabolism Measuring exponents ◮ hierarchical network History: River networks Earlier theories ◮ capillaries (delivery units) invariant Geometric argument Blood networks River networks ◮ network impedance is minimized via evolution Conclusion References Claims: ◮ P ∝ M 3 / 4 ◮ networks are fractal ◮ quarter powers everywhere Frame 84/117
Scaling Impedance measures: Scaling-at-large Allometry Definitions Poiseuille flow (outer branches): Examples History: Metabolism Measuring exponents History: River networks N Z = 8 µ ℓ k Earlier theories � Geometric argument π r 4 Blood networks k N k River networks k = 0 Conclusion References Pulsatile flow (main branches): N h 1 / 2 � k Z ∝ r 5 / 2 N k k = 0 k Frame 85/117
Scaling Not so fast . . . Scaling-at-large Allometry Actually, model shows: Definitions Examples ◮ P ∝ M 3 / 4 does not follow for pulsatile flow History: Metabolism Measuring exponents History: River networks ◮ networks are not necessarily fractal. Earlier theories Geometric argument Blood networks River networks Conclusion Do find: References ◮ Murray’s cube law (1927) for outer branches: r 3 0 = r 3 1 + r 3 2 ◮ Impedance is distributed evenly. ◮ Can still assume networks are fractal. Frame 86/117
Scaling Connecting network structure to α 1. Ratios of network parameters: Scaling-at-large R n = n k + 1 , R ℓ = ℓ k + 1 , R r = r k + 1 Allometry n k ℓ k r k Definitions Examples 2. Number of capillaries ∝ P ∝ M α . History: Metabolism Measuring exponents History: River networks Earlier theories α = − ln R n Geometric argument ⇒ Blood networks ln R 2 r R ℓ River networks Conclusion (also problematic due to prefactor issues) References Soldiering on, assert: ◮ area-preservingness: R r = R − 1 / 2 n ◮ space-fillingness: R ℓ = R − 1 / 3 n ◮ ⇒ α = 3 / 4 Frame 87/117
Scaling Data from real networks R − 1 R − 1 − ln R r − ln R ℓ Scaling-at-large Network R n α r ℓ ln R n ln R n Allometry Definitions West et al. – – – 1/2 1/3 3/4 Examples History: Metabolism Measuring exponents History: River networks rat (PAT) 2.76 1.58 1.60 0.45 0.46 0.73 Earlier theories Geometric argument Blood networks cat (PAT) 3.67 1.71 1.78 0.41 0.44 0.79 River networks Conclusion (Turcotte et al. [24] ) References dog (PAT) 3.69 1.67 1.52 0.39 0.32 0.90 pig (LCX) 3.57 1.89 2.20 0.50 0.62 0.62 pig (RCA) 3.50 1.81 2.12 0.47 0.60 0.65 pig (LAD) 3.51 1.84 2.02 0.49 0.56 0.65 human (PAT) 3.03 1.60 1.49 0.42 0.36 0.83 human (PAT) 3.36 1.56 1.49 0.37 0.33 0.94 Frame 88/117
Scaling Simple supply networks Scaling-at-large Allometry ◮ Banavar et al., Definitions Examples Nature, History: Metabolism Measuring exponents (1999) [1] History: River networks Earlier theories ◮ Flow rate Geometric argument Blood networks argument River networks Conclusion ◮ Ignore References impedance ◮ Very general attempt to find most efficient transportation networks Frame 89/117
Scaling Simple supply networks Scaling-at-large ◮ Banavar et al. find ‘most efficient’ networks with Allometry Definitions Examples History: Metabolism P ∝ M d / ( d + 1 ) Measuring exponents History: River networks Earlier theories ◮ ... but also find Geometric argument Blood networks River networks Conclusion V network ∝ M ( d + 1 ) / d References ◮ d = 3: V blood ∝ M 4 / 3 ◮ Consider a 3 g shrew with V blood = 0 . 1 V body ◮ ⇒ 3000 kg elephant with V blood = 10 V body ◮ Such a pachyderm would be rather miserable. Frame 90/117
Scaling Geometric argument ◮ Consider one source supplying many sinks in a Scaling-at-large d -dim. volume in a D -dim. ambient space. Allometry Definitions ◮ Assume sinks are invariant. Examples History: Metabolism ◮ Assume ρ = ρ ( V ) . Measuring exponents History: River networks Earlier theories ◮ Assume some cap on flow speed of material. Geometric argument Blood networks ◮ See network as a bundle of virtual vessels: River networks Conclusion References ◮ Q: how does the number of sustainable sinks N sinks scale with volume V for the most efficient network design? ◮ Or: what is the highest α for N sinks ∝ V α ? Frame 92/117
Scaling Geometric argument ◮ Allometrically growing regions: Scaling-at-large Allometry Definitions Examples History: Metabolism L Ω L’ (V’) Measuring exponents 2 Ω 2 (V) History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion L 1 L’ 1 References ◮ Have d length scales which scale as L i ∝ V γ i where γ 1 + γ 2 + . . . + γ d = 1. ◮ For isometric growth, γ i = 1 / d . ◮ For allometric growth, we must have at least two of the { γ i } being different Frame 93/117
Scaling Geometric argument Scaling-at-large Allometry ◮ Best and worst configurations (Banavar et al.) Definitions Examples History: Metabolism a b Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References ◮ Rather obviously: min V net ∝ � distances from source to sinks. Frame 94/117
Scaling Minimal network volume: Scaling-at-large Allometry Real supply networks are close to optimal: Definitions Examples (a) (b) (c) (d) History: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Figure 1. (a) Commuter rail network in the Boston area. The arrow marks the assumed root of the network. (b) Star graph. (c) Minimum spanning tree. (d) The model of equation (3) applied to the same set of stations. (2006) Gastner and Newman [10] : “Shape and efficiency in spatial distribution networks” Frame 95/117
Scaling Minimal network volume: Scaling-at-large Allometry Definitions Examples Approximate network volume by integral over region: History: Metabolism Measuring exponents History: River networks � Earlier theories ρ || � x || d � min V net ∝ x Geometric argument Blood networks Ω d , D ( V ) River networks Conclusion � References → ρ V 1 + γ max ( c 2 1 u 2 1 + . . . + c 2 k u 2 k ) 1 / 2 d � u Ω d , D ( c ) ∝ ρ V 1 + γ max Frame 96/117
Scaling Geometric argument ◮ General result: Scaling-at-large min V net ∝ ρ V 1 + γ max Allometry Definitions Examples History: Metabolism Measuring exponents History: River networks Earlier theories ◮ If scaling is isometric, we have γ max = 1 / d : Geometric argument Blood networks min V net / iso ∝ ρ V 1 + 1 / d = ρ V ( d + 1 ) / d River networks Conclusion References ◮ If scaling is allometric, we have γ max = γ allo > 1 / d : and min V net / allo ∝ ρ V 1 + γ allo ◮ Isometrically growing volumes require less network volume than allometrically growing volumes: min V net / iso → 0 as V → ∞ min V net / allo Frame 97/117
Scaling Blood networks Scaling-at-large Allometry Definitions ◮ Material costly ⇒ expect lower optimal bound of Examples V net ∝ ρ V ( d + 1 ) / d to be followed closely. History: Metabolism Measuring exponents History: River networks ◮ For cardiovascular networks, d = D = 3. Earlier theories Geometric argument ◮ Blood volume scales linearly with body volume [23] , Blood networks River networks Conclusion V net ∝ V . References ◮ Sink density must ∴ decrease as volume increases: ρ ∝ V − 1 / d . ◮ Density of suppliable sinks decreases with organism size. Frame 99/117
Scaling Blood networks Scaling-at-large Allometry Definitions ◮ Then P , the rate of overall energy use in Ω , can at Examples History: Metabolism most scale with volume as Measuring exponents History: River networks Earlier theories P ∝ ρ V ∝ ρ M ∝ M ( d − 1 ) / d Geometric argument Blood networks River networks Conclusion References ◮ For d = 3 dimensional organisms, we have P ∝ M 2 / 3 Frame 100/117
Scaling Recap: Scaling-at-large Allometry Definitions Examples History: Metabolism Measuring exponents ◮ The exponent α = 2 / 3 works for all birds and History: River networks Earlier theories mammals up to 10–30 kg Geometric argument Blood networks ◮ For mammals > 10–30 kg, maybe we have a new River networks Conclusion scaling regime References ◮ Economos: limb length break in scaling around 20 kg ◮ White and Seymour, 2005: unhappy with large herbivore measurements. Find α ≃ 0 . 686 ± 0 . 014 Frame 101/117
Scaling Prefactor: Scaling-at-large Stefan-Boltzmann law: Allometry Definitions ◮ Examples d E History: Metabolism d t = σ ST 4 Measuring exponents History: River networks Earlier theories Geometric argument where S is surface and T is temperature. Blood networks River networks ◮ Very rough estimate of prefactor based on scaling of Conclusion normal mammalian body temperature and surface References area S : B ≃ 10 5 M 2 / 3 erg/sec . ◮ Measured for M ≤ 10 kg: B = 2 . 57 × 10 5 M 2 / 3 erg/sec . Frame 102/117
Scaling River networks Scaling-at-large Allometry ◮ View river networks as collection networks. Definitions Examples ◮ Many sources and one sink. History: Metabolism Measuring exponents ◮ Assume ρ is constant over time: History: River networks Earlier theories Geometric argument Blood networks V net ∝ ρ V ( d + 1 ) / d = constant × V 3 / 2 River networks Conclusion References ◮ Network volume grows faster than basin ‘volume’ (really area). ◮ It’s all okay: Landscapes are d =2 surfaces living in D =3 dimension. ◮ Streams can grow not just in width but in depth... Frame 104/117
Scaling Hack’s law ◮ Volume of water in river network can be calculated by Scaling-at-large adding up basin areas Allometry ◮ Flows sum in such a way that Definitions Examples History: Metabolism Measuring exponents � V net = a pixel i History: River networks Earlier theories all pixels Geometric argument Blood networks River networks ◮ Hack’s law again: Conclusion ℓ ∼ a h References ◮ Can argue V net ∝ V 1 + h basin = a 1 + h basin where h is Hack’s exponent. ◮ ∴ minimal volume calculations gives h = 1 / 2 Frame 105/117
Scaling Real data: Scaling-at-large Allometry Definitions ◮ Banavar et al.’s Examples approach [1] is History: Metabolism Measuring exponents History: River networks okay because ρ Earlier theories Geometric argument really is constant. Blood networks River networks ◮ The irony: shows Conclusion References optimal basins are isometric ◮ Optimal Hack’s law: ℓ ∼ a h with h = 1 / 2 ◮ (Zzzzz) From Banavar et al. (1999) [1] Frame 106/117
Scaling Even better—prefactors match up: 20 Scaling-at-large Amazon 19 Allometry Mississippi Definitions 18 Examples Congo log 10 water volume V [m 3 ] History: Metabolism Measuring exponents 17 Nile History: River networks Earlier theories 16 Geometric argument Blood networks 15 River networks Conclusion 14 References 13 12 11 10 9 8 6 7 8 9 10 11 12 13 log 10 area a [m 2 ] Frame 107/117
Scaling Conclusion Scaling-at-large Allometry ◮ Supply network story consistent with dimensional Definitions Examples analysis. History: Metabolism Measuring exponents ◮ Isometrically growing regions can be more efficiently History: River networks Earlier theories supplied than allometrically growing ones. Geometric argument Blood networks River networks ◮ Ambient and region dimensions matter Conclusion ( D = d versus D > d ). References ◮ Deviations from optimal scaling suggest inefficiency (e.g., gravity for organisms, geological boundaries). ◮ Actual details of branching networks not that important. ◮ Exact nature of self-similarity varies. Frame 109/117
Scaling References I Scaling-at-large J. R. Banavar, A. Maritan, and A. Rinaldo. Allometry Definitions Size and form in efficient transportation networks. Examples History: Metabolism Nature , 399:130–132, 1999. pdf ( ⊞ ) Measuring exponents History: River networks Earlier theories P . Bennett and P . Harvey. Geometric argument Blood networks Active and resting metabolism in birds—allometry, River networks Conclusion phylogeny and ecology. References J. Zool. , 213:327–363, 1987. L. M. A. Bettencourt, J. Lobo, D. Helbing, Kühnhert, and G. B. West. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl. Acad. Sci. , 104(17):7301–7306, 2007. pdf ( ⊞ ) Frame 110/117
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