Scalinga Plenitude of Power Laws Scaling-at-large Principles of - PowerPoint PPT Presentation

Scaling Non-uniform growth—arm length versus height: Scaling-at-large Allometry Examples A focus: Metabolism Good example of a break in scaling: 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 [28] 26 of 124

Scaling Weightlifting: M worldrecord ∝ M 2 / 3 lifter Scaling-at-large Allometry Examples A focus: 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 [28] 27 of 124

Scaling Titanothere horns: L horn ∼ L 4 skull Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References p. 36, McMahon and Bonner [28] ; a bit dubious. 28 of 124

Scaling The allometry of nails: Observed: Diameter ∝ Length 2 / 3 or d ∝ ℓ 2 / 3 . Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References Since ℓ d 2 ∝ Volume v : ◮ Diameter ∝ Mass 3 / 8 or d ∝ v 3 / 8 . ◮ Length ∝ Mass 1 / 4 or ℓ ∝ v 1 / 4 . ◮ Nails lengthen faster than they broaden (c.f. trees). p. 58–59, McMahon and Bonner [28] 29 of 124

Scaling The allometry of nails: Scaling-at-large Allometry A buckling instability?: Examples A focus: Metabolism ◮ Physics/Engineering result ( ⊞ ): Columns buckle Measuring exponents History: River networks under a load which depends on d 4 /ℓ 2 . Earlier theories Geometric argument Blood networks ◮ To drive nails in, resistive force ∝ nail circumference River networks Conclusion = π d . References ◮ Match forces independent of nail size: d 4 /ℓ 2 ∝ d . ◮ Leads to d ∝ ℓ 2 / 3 . ◮ Argument made by Galileo [14] in 1638 in “Discourses on Two New Sciences.” ( ⊞ ) Also, see here. ( ⊞ ) ◮ Euler, 1757. ( ⊞ ) ◮ Also see McMahon, “Size and Shape in Biology,” Science, 1973. [26] 30 of 124

Scaling Rowing: Speed ∝ (number of rowers) 1 / 9 Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References 31 of 124

Scaling From further back: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument ◮ Zipf (more later) Blood networks River networks Conclusion ◮ Survey by Naroll and von Bertalanffy [31] References “The principle of allometry in biology and the social sciences” General Systems, Vol 1., 1956. 32 of 124

Scaling Scaling in Cities: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents ◮ “Growth, innovation, scaling, and the pace of life in History: River networks Earlier theories cities” Geometric argument Blood networks Bettencourt et al., PNAS, 2007. [4] River networks Conclusion ◮ Quantified levels of References ◮ Infrastructure ◮ Wealth ◮ Crime levels ◮ Disease ◮ Energy consumption as a function of city size N (population). 33 of 124

Scaling Scaling in Cities: Scaling-at-large Table 1. Scaling exponents for urban indicators vs. city size Allometry Y 95% CI Adj- R 2 Observations Country–year � Examples A focus: Metabolism New patents 1.27 � 1.25,1.29 � 0.72 331 U.S. 2001 Measuring exponents Inventors 1.25 � 1.22,1.27 � 0.76 331 U.S. 2001 History: River networks Private R&D employment 1.34 � 1.29,1.39 � 0.92 266 U.S. 2002 Earlier theories � Supercreative � employment 1.15 � 1.11,1.18 � 0.89 287 U.S. 2003 Geometric argument R&D establishments 1.19 � 1.14,1.22 � 0.77 287 U.S. 1997 Blood networks R&D employment 1.26 � 1.18,1.43 � 0.93 295 China 2002 River networks Total wages 1.12 � 1.09,1.13 � 0.96 361 U.S. 2002 Conclusion Total bank deposits 1.08 � 1.03,1.11 � 0.91 267 U.S. 1996 References GDP 1.15 � 1.06,1.23 � 0.96 295 China 2002 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 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. 34 of 124

Scaling Scaling in Cities: Scaling-at-large Intriguing findings: Allometry Examples A focus: Metabolism ◮ Global supply costs scale sublinearly with N ( β < 1). Measuring exponents History: River networks ◮ Returns to scale for infrastructure. Earlier theories Geometric argument ◮ Total individual costs scale linearly with N ( β = 1) Blood networks River networks ◮ Individuals consume similar amounts independent of Conclusion city size. References ◮ Social quantities scale superlinearly with N ( β > 1) ◮ 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. 35 of 124

Scaling Ecology—Species-area law: ( ⊞ ) Scaling-at-large Allometry Examples A focus: Metabolism Allegedly (data is messy): Measuring exponents History: River networks Earlier theories ◮ Geometric argument N species ∝ A β Blood networks River networks Conclusion ◮ On islands: β ≈ 1 / 4. References ◮ On continuous land: β ≈ 1 / 8. A focus: ◮ How much energy do organisms need to live? ◮ And how does this scale with organismal size? 36 of 124

Scaling Animal power Scaling-at-large Allometry Examples A focus: Metabolism Fundamental biological and ecological constraint: Measuring exponents History: River networks Earlier theories P = c M α Geometric argument Blood networks River networks P = basal metabolic rate Conclusion References M = organismal body mass 38 of 124

P = c M α Scaling Scaling-at-large Allometry Prefactor c depends on body plan and body temperature: Examples A focus: Metabolism Measuring exponents 39–41 ◦ C Birds History: River networks Earlier theories 36–38 ◦ C Eutherian Mammals Geometric argument Blood networks 34–36 ◦ C Marsupials River networks Conclusion 30–31 ◦ C Monotremes References 39 of 124

Scaling What one might expect: Scaling-at-large Allometry Examples α = 2 / 3 because . . . A focus: Metabolism Measuring exponents ◮ Dimensional analysis suggests History: River networks Earlier theories an energy balance surface law: Geometric argument Blood networks River networks P ∝ S ∝ V 2 / 3 ∝ M 2 / 3 Conclusion References ◮ Lognormal fluctuations: Gaussian fluctuations in log P around log cM α . ◮ Stefan-Boltzmann law ( ⊞ ) for radiated energy: d E d t = σε ST 4 ∝ S 40 of 124

Scaling The prevailing belief of the church of quarterology Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks α = 3 / 4 Conclusion References P ∝ M 3 / 4 Huh? 41 of 124

Scaling The prevailing belief of the church of quarterology Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Most obvious concern: Geometric argument Blood networks River networks 3 / 4 − 2 / 3 = 1 / 12 Conclusion References ◮ An exponent higher than 2/3 points suggests a fundamental inefficiency in biology. ◮ Organisms must somehow be running ‘hotter’ than they need to balance heat loss. 42 of 124

Scaling Related putative scalings: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Wait! There’s more!: Geometric argument Blood networks ◮ number of capillaries ∝ M 3 / 4 River networks Conclusion ◮ time to reproductive maturity ∝ M 1 / 4 References ◮ heart rate ∝ M − 1 / 4 ◮ cross-sectional area of aorta ∝ M 3 / 4 ◮ population density ∝ M − 3 / 4 43 of 124

Scaling The great ‘law’ of heartbeats: Scaling-at-large Assuming: Allometry Examples A focus: Metabolism ◮ Average lifespan ∝ M β Measuring exponents History: River networks ◮ Average heart rate ∝ M − β Earlier theories Geometric argument Blood networks ◮ Irrelevant but perhaps β = 1 / 4. River networks Conclusion References Then: ◮ 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.... 44 of 124

Scaling History Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents 1840’s: Sarrus and Rameaux [36] first suggested α = 2 / 3. History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References 45 of 124

Scaling History Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents 1883: Rubner [34] found α ≃ 2 / 3. History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References 46 of 124

Scaling History Scaling-at-large Allometry Examples A focus: Metabolism 1930’s: Brody, Benedict study mammals. [7] Measuring exponents History: River networks Earlier theories Found α ≃ 0 . 73 (standard). Geometric argument Blood networks River networks Conclusion References 47 of 124

Scaling History Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References ◮ 1932: Kleiber analyzed 13 mammals. [23] ◮ Found α = 0 . 76 and suggested α = 3 / 4. ◮ Scaling law of Metabolism became known as Kleiber’s Law ( ⊞ ) (2011 Wikipedia entry is embarrassing). ◮ 1961 book: “The Fire of Life. An Introduction to Animal Energetics”. [24] 48 of 124

Scaling History Scaling-at-large Allometry Examples 1950/1960: Hemmingsen [19, 20] A focus: Metabolism Measuring exponents History: River networks Extension to unicellular organisms. Earlier theories Geometric argument α = 3 / 4 assumed true. Blood networks River networks Conclusion References 49 of 124

Scaling History Scaling-at-large Allometry Examples 1964: Troon, Scotland: [5] A focus: Metabolism Measuring exponents History: River networks 3rd symposium on energy metabolism. Earlier theories Geometric argument α = 3 / 4 made official . . . . . . 29 to zip. Blood networks River networks Conclusion References 50 of 124

Scaling Today Scaling-at-large Allometry ◮ 3/4 is held by many to be the one true exponent. Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks In the Beat of a Heart: Life, Energy, and Conclusion References the Unity of Nature —by John Whitfield ◮ But—much controversy... ◮ See ‘Re-examination of the “3/4-law” of metabolism’ Dodds, Rothman, and Weitz [12] and ensuing madness... 51 of 124

Scaling Some data on metabolic rates Scaling-at-large Allometry Examples 3.5 A focus: Metabolism B = 0.026 M 0.668 Measuring exponents 3 History: River networks Earlier theories 2.5 Geometric argument ◮ Heusner’s Blood networks 2 River networks data Conclusion 1.5 [source=/home/dodds/work/biology/allometry/heusner/figures/figheusner391.ps] log 10 B (1991) [21] References 1 ◮ 391 Mammals 0.5 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 52 of 124

Scaling Some data on metabolic rates Scaling-at-large 2 Allometry Examples B = 0.041 M 0.664 A focus: Metabolism 1.5 Measuring exponents History: River networks Earlier theories ◮ Bennett and 1 Geometric argument Blood networks Harvey’s data River networks log 10 B 0.5 Conclusion (1987) [3] References 0 ◮ 398 birds ◮ blue line: 2/3 −0.5 ◮ red line: 3/4. −1 ◮ ( B = P ) −1.5 0 1 2 3 4 5 log 10 M ◮ Passerine vs. non-passerine issue... 53 of 124

Scaling Linear regression Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents Important: History: River networks Earlier theories Geometric argument ◮ Ordinary Least Squares (OLS) Linear regression is Blood networks River networks only appropriate for analyzing a dataset { ( x i , y i ) } Conclusion References when we know the x i are measured without error. ◮ Here we assume that measurements of mass M have less error than measurements of metabolic rate B . ◮ Linear regression assumes Gaussian errors. 55 of 124

Scaling Measuring exponents Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks More on regression: Earlier theories Geometric argument Blood networks If (a) we don’t know what the errors of either variable are, River networks Conclusion or (b) no variable can be considered independent, References then we need to use Standardized Major Axis Linear Regression. [35, 33] (aka Reduced Major Axis = RMA.) 56 of 124

Scaling Measuring exponents Scaling-at-large Allometry Examples A focus: 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! ◮ Scale invariant. 57 of 124

Scaling Measuring exponents Scaling-at-large Allometry Examples A focus: Metabolism Relationship to ordinary least squares regression is Measuring exponents History: River networks simple: Earlier theories Geometric argument Blood networks River networks r − 1 × slope OLS y on x Conclusion = slope SMA References = r × slope OLS x on y 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 58 of 124

Scaling Heusner’s data, 1991 (391 Mammals) Scaling-at-large range of M N α ˆ Allometry Examples A focus: Metabolism Measuring exponents ≤ 0 . 1 kg 167 0 . 678 ± 0 . 038 History: River networks Earlier theories Geometric argument Blood networks ≤ 1 kg 0 . 662 ± 0 . 032 276 River networks Conclusion References ≤ 10 kg 357 0 . 668 ± 0 . 019 ≤ 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 59 of 124

Scaling Bennett and Harvey, 1987 (398 birds) α ˆ M max N Scaling-at-large Allometry Examples ≤ 0 . 032 162 0 . 636 ± 0 . 103 A focus: Metabolism Measuring exponents History: River networks Earlier theories ≤ 0 . 1 236 0 . 602 ± 0 . 060 Geometric argument Blood networks River networks Conclusion ≤ 0 . 32 290 0 . 607 ± 0 . 039 References ≤ 1 334 0 . 652 ± 0 . 030 ≤ 3 . 2 371 0 . 655 ± 0 . 023 ≤ 10 0 . 664 ± 0 . 020 391 ≤ 32 396 0 . 665 ± 0 . 019 ≤ 100 398 0 . 664 ± 0 . 019 60 of 124

Scaling Hypothesis testing Scaling-at-large Allometry Test to see if α ′ is consistent with our data { ( M i , B i ) } : Examples A focus: Metabolism Measuring exponents H 0 : α = α ′ and H 1 : α � = α ′ . History: River networks Earlier theories Geometric argument Blood networks River networks ◮ Assume each B i (now a random variable) is normally Conclusion distributed about α ′ log 10 M i + log 10 c . References ◮ Follows that the measured α for one realization 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.” [9] 61 of 124

Scaling Revisiting the past—mammals Scaling-at-large Allometry Examples Full mass range: A focus: Metabolism Measuring exponents History: River networks N α ˆ p 2 / 3 p 3 / 4 Earlier theories Geometric argument Blood networks River networks < 10 − 6 Kleiber 13 0.738 0.11 Conclusion References < 10 − 4 < 10 − 2 Brody 35 0.718 < 10 − 6 < 10 − 5 Heusner 391 0.710 < 10 − 15 Bennett 398 0.664 0.69 and Harvey 62 of 124

Scaling Revisiting the past—mammals M ≤ 10 kg: Scaling-at-large Allometry N α ˆ p 2 / 3 p 3 / 4 Examples A focus: Metabolism Measuring exponents History: River networks Kleiber 5 0.667 0.99 0.088 Earlier theories Geometric argument Blood networks < 10 − 3 < 10 − 3 Brody 26 0.709 River networks Conclusion References < 10 − 15 Heusner 357 0.668 0.91 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 63 of 124

Scaling Fluctuations—Things look normal... Scaling-at-large [07−Nov−1999 peter dodds] 4 Allometry Examples 20 bins A focus: Metabolism 3.5 Measuring exponents History: River networks 3 Earlier theories P( log 10 B / M 2/3 ) Geometric argument Blood networks [source=/home/dodds/work/biology/allometry/heusner/figures/figmetascalingfn2.ps] 2.5 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 ) ◮ Use a Kolmogorov-Smirnov test. 64 of 124

Scaling Analysis of residuals Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents 1. Presume an exponent of your choice: 2/3 or 3/4. History: River networks Earlier theories 2. Fit the prefactor (log 10 c ) and then examine the Geometric argument Blood networks residuals: River networks Conclusion References r i = log 10 B i − ( α ′ log 10 M i − log 10 c ) . 3. H 0 : residuals are uncorrelated H 1 : residuals are correlated. 4. Measure the correlations in the residuals and compute a p -value. 65 of 124

Scaling Analysis of residuals Scaling-at-large Allometry We use the spiffing Spearman Rank-Order Correlation Examples A focus: Metabolism Cofficient ( ⊞ ) Measuring exponents History: River networks Earlier theories Geometric argument Basic idea: Blood networks River networks Conclusion ◮ Given { ( x i , y i ) } , rank the { x i } and { y i } separately References from smallest to largest. Call these ranks R i and S i . ◮ Now calculate correlation coefficient for ranks, r s : ◮ � n i = 1 ( R i − ¯ R )( S i − ¯ S ) r s = �� n �� n i = 1 ( S i − ¯ i = 1 ( R i − ¯ R ) 2 S ) 2 ◮ Perfect correlation: x i ’s and y i ’s both increase monotonically. 66 of 124

Scaling Analysis of residuals Scaling-at-large Allometry Examples A focus: Metabolism We assume all rank orderings are equally likely: Measuring exponents History: River networks Earlier theories ◮ r s is distributed according to a Student’s Geometric argument Blood networks t -distribution ( ⊞ ) with N − 2 degrees of freedom. River networks Conclusion ◮ Excellent feature: Non-parametric—real distribution References of x ’s and y ’s doesn’t matter. ◮ Bonus: works for non-linear monotonic relationships as well. ◮ See Numerical Recipes in C/Fortran ( ⊞ ) which contains many good things. [32] 67 of 124

Scaling Analysis of residuals—mammals Scaling-at-large Allometry 0 0 Examples (a) (b) A focus: Metabolism −1 −1 Measuring exponents History: River networks −2 −2 Earlier theories Geometric argument Blood networks −3 −3 River networks Conclusion (a) M < 3 . 2 kg, −4 −4 log 10 p References 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 (b) M < 10 kg, 0 0 (c) M < 32 kg, (c) (d) −1 −1 (d) all −2 −2 mammals. −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 ’ 68 of 124

Scaling Analysis of residuals—birds Scaling-at-large Allometry 0 0 Examples (a) (b) A focus: Metabolism −1 −1 Measuring exponents History: River networks −2 −2 Earlier theories Geometric argument Blood networks −3 −3 River networks Conclusion −4 −4 log 10 p (a) M < 0 . 1 kg, References 0.6 2/3 0.7 3/4 0.8 0.6 2/3 0.7 3/4 0.8 (b) M < 1 kg, 0 0 (c) (d) (c) M < 10 kg, −1 −1 (d) all birds. −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 ’ 69 of 124

Scaling Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents Other approaches to measuring exponents: History: River networks Earlier theories Geometric argument ◮ Clauset, Shalizi, Newman: “Power-law distributions Blood networks River networks in empirical data” [8] Conclusion References SIAM Review, 2009. ◮ See Clauset’s page on measuring power law exponents ( ⊞ ) (code, other goodies). 70 of 124

Scaling Recap: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks ◮ So: The exponent α = 2 / 3 works for all birds and Earlier theories Geometric argument mammals up to 10–30 kg Blood networks River networks ◮ For mammals > 10–30 kg, maybe we have a new Conclusion References scaling regime ◮ Possible connection?: Economos (1983)—limb length break in scaling around 20 kg [13] ◮ But see later: non-isometric growth leads to lower metabolic scaling. Oops. 71 of 124

Scaling The widening gyre: Scaling-at-large Now we’re really confused (empirically): Allometry Examples A focus: Metabolism ◮ White and Seymour, 2005: unhappy with large Measuring exponents History: River networks herbivore measurements [43] . Pro 2 / 3: Find Earlier theories Geometric argument α ≃ 0 . 686 ± 0 . 014. Blood networks River networks ◮ Glazier, BioScience (2006) [17] : “The 3/4-Power Law Conclusion References Is Not Universal: Evolution of Isometric, Ontogenetic Metabolic Scaling in Pelagic Animals.” ◮ Glazier, Biol. Rev. (2005) [16] : “Beyond the 3/4-power law’: variation in the intra- and interspecific scaling of metabolic rate in animals.” ◮ Savage et al., PLoS Biology (2008) [37] “Sizing up allometric scaling theory” Pro 3 / 4: problems claimed to be finite-size scaling. 72 of 124

0 L l ? 0 l Scaling Basic basin quantities: a , l , L � , L ⊥ : 0 a 0 L a k Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument ◮ a = drainage Blood networks L = L k River networks basin area Conclusion References ◮ ℓ = length of L ? longest (main) stream ◮ L = L � = longitudinal length of basin 74 of 124

Scaling River networks Scaling-at-large Allometry Examples A focus: Metabolism ◮ 1957: J. T. Hack [18] Measuring exponents History: River networks Earlier theories “Studies of Longitudinal Stream Profiles in Virginia Geometric argument Blood networks and Maryland” River networks ℓ ∼ a h Conclusion References h ∼ 0 . 6 ◮ 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. 75 of 124

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 right area four 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 We cannot compare our data more quanti- relation is well approximated by the simple, 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- low-order channel networks, at channel derived from our data and those reported works (11-15) provides a composite data heads, and along unchanneled valleys in previously reflects variation in the head- set that also is reasonably fit (5) by this 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 defined as the length along the main valley the world's largest rivers (Fig. 2). This 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 Scaling Large-scale networks: which the landscape is dissected (Fig. 3). rivers to the finest scale valleys. Within this In the field this scale is easily recognized as L = 1.78 A49 (1) (1992) Montgomery and Dietrich [29] : scale range there appears to be little inher- Scaling-at-large ent to the channel network and to the Allometry Examples corresponding shape of the drainage area it A focus: Metabolism Measuring exponents captures that provides reference to an ab- History: River networks Earlier theories solute scale. E Fig. 2. Basin length versus drainage Geometric argument Nonetheless, field studies in semiarid to Blood networks area for unchanneled valleys, source River networks humid regions demonstrate that there is a areas, and low-order channels mapped Conclusion 5 this study (0) and mainstream in finite extent to the branching channel net- References U length versus drainage area data report- work (4, 5, 19-22). Channels do not occupy c ed for large channel networks (0). 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) 827 REPORT ◮ 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. 76 of 124

Scaling World’s largest rivers only: Scaling-at-large Allometry 4 Examples 10 A focus: Metabolism (mi) Measuring exponents History: River networks Earlier theories l length Geometric argument Blood networks 3 River networks 10 stream Conclusion area a (sq mi) References main 2 10 4 5 6 7 10 10 10 10 ◮ Data from Leopold (1994) [25, 11] ◮ Estimate of Hack exponent: h = 0 . 50 ± 0 . 06 77 of 124

Scaling Earlier theories Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents Building on the surface area idea... History: River networks Earlier theories ◮ Blum (1977) [6] speculates on four-dimensional Geometric argument Blood networks River networks biology: Conclusion P ∝ M ( d − 1 ) / d References ◮ d = 3 gives α = 2 / 3 ◮ d = 4 gives α = 3 / 4 ◮ So we need another dimension... ◮ Obviously, a bit silly. . . [39] 79 of 124

Scaling Earlier theories Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Building on the surface area idea: Geometric argument Blood networks ◮ McMahon (70’s, 80’s): Elastic Similarity [26, 28] River networks Conclusion ◮ Idea is that organismal shapes scale allometrically References with 1/4 powers (like trees...) ◮ Appears to be true for ungulate legs... [27] ◮ Metabolism and shape never properly connected. 80 of 124

Scaling Nutrient delivering networks: ◮ 1960’s: Rashevsky considers blood networks and Scaling-at-large finds a 2 / 3 scaling. Allometry Examples ◮ 1997: West et al. [42] use a network story to find 3 / 4 A focus: Metabolism Measuring exponents scaling. History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References 81 of 124

Scaling Nutrient delivering networks: Scaling-at-large Allometry Examples A focus: Metabolism West et al.’s assumptions: Measuring exponents History: River networks Earlier theories 1. hierarchical network Geometric argument Blood networks 2. capillaries (delivery units) invariant River networks Conclusion 3. network impedance is minimized via evolution References Claims: ◮ P ∝ M 3 / 4 ◮ networks are fractal ◮ quarter powers everywhere 82 of 124

Scaling Impedance measures: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents Poiseuille flow (outer branches): History: River networks Earlier theories Geometric argument N Blood networks Z = 8 µ ℓ k River networks � Conclusion π r 4 k N k References k = 0 Pulsatile flow (main branches): N h 1 / 2 � k Z ∝ r 5 / 2 N k k = 0 k 83 of 124

Scaling Not so fast . . . Scaling-at-large Allometry Examples Actually, model shows: A focus: Metabolism Measuring exponents ◮ P ∝ M 3 / 4 does not follow for pulsatile flow History: River networks Earlier theories Geometric argument ◮ networks are not necessarily fractal. Blood networks River networks Conclusion References Do find: ◮ Murray’s cube law (1927) for outer branches: [30] r 3 0 = r 3 1 + r 3 2 ◮ Impedance is distributed evenly. ◮ Can still assume networks are fractal. 84 of 124

Scaling Connecting network structure to α 1. Ratios of network parameters: Scaling-at-large Allometry R n = n k + 1 , R ℓ = ℓ k + 1 , R r = r k + 1 Examples A focus: Metabolism n k ℓ k r k Measuring exponents History: River networks Earlier theories 2. Number of capillaries ∝ P ∝ M α . Geometric argument Blood networks River networks Conclusion α = − ln R n References ⇒ ln R 2 r R ℓ (also problematic due to prefactor issues) Soldiering on, assert: ◮ area-preservingness: R r = R − 1 / 2 n ◮ space-fillingness: R ℓ = R − 1 / 3 n ◮ ⇒ α = 3 / 4 85 of 124

Scaling Data from real networks Scaling-at-large R − 1 R − 1 − ln R r − ln R ℓ Network R n α r Allometry ℓ ln R n ln R n Examples A focus: Metabolism West et al. – – – 1/2 1/3 3/4 Measuring exponents History: River networks Earlier theories Geometric argument rat (PAT) 2.76 1.58 1.60 0.45 0.46 0.73 Blood networks River networks Conclusion cat (PAT) 3.67 1.71 1.78 0.41 0.44 0.79 References (Turcotte et al. [41] ) 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 86 of 124

Scaling Really, quite confused: Whole 2004 issue of Functional Ecology addresses Scaling-at-large Allometry the problem: Examples A focus: Metabolism ◮ J. Kozlowski, M. Konrzewski (2004). “Is West, Brown Measuring exponents History: River networks Earlier theories and Enquist’s model of allometric scaling Geometric argument Blood networks mathematically correct and biologically relevant?” River networks Conclusion Functional Ecology 18: 283–9, 2004. References ◮ J. H. Brown, G. B. West, and B. J. Enquist. “Yes, West, Brown and Enquist’s model of allometric scaling is both mathematically correct and biologically relevant.” Functional Ecology 19: 735–738, 2005. ◮ J. Kozlowski, M. Konarzewski (2005). “West, Brown and Enquist’s model of allometric scaling again: the same questions remain.” Functional Ecology 19: 739–743, 2005. 87 of 124

Scaling Simple supply networks Scaling-at-large Allometry Examples ◮ Banavar et al., A focus: Metabolism Measuring exponents Nature, History: River networks Earlier theories (1999) [1] Geometric argument Blood networks River networks ◮ Flow rate Conclusion argument References ◮ Ignore impedance ◮ Very general attempt to find most efficient transportation networks 88 of 124

Scaling Simple supply networks Scaling-at-large Allometry Examples ◮ Banavar et al. find ‘most efficient’ networks with A focus: Metabolism Measuring exponents History: River networks P ∝ M d / ( d + 1 ) Earlier theories Geometric argument Blood networks River networks ◮ ... but also find Conclusion References V network ∝ M ( d + 1 ) / d ◮ 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 89 of 124

Scaling Simple supply networks Scaling-at-large Allometry Examples A focus: Metabolism Such a pachyderm would be rather miserable: Measuring exponents History: River networks Earlier theories Geometric argument Blood networks River networks Conclusion References 90 of 124

Scaling Geometric argument Scaling-at-large ◮ “Optimal Form of Branching Supply and Collection Allometry Examples Networks.” Dodds, Phys. Rev. Lett., 2010. [10] A focus: Metabolism Measuring exponents History: River networks ◮ Consider one source supplying many sinks in a Earlier theories Geometric argument d -dim. volume in a D -dim. ambient space. Blood networks River networks ◮ Assume sinks are invariant. Conclusion References ◮ Assume sink density ρ = ρ ( V ) . ◮ Assume some cap on flow speed of material. ◮ See network as a bundle of virtual vessels: 92 of 124

Scaling Geometric argument Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks ◮ Q: how does the number of sustainable sinks N sinks River networks Conclusion scale with volume V for the most efficient network References design? ◮ Or: what is the highest α for N sinks ∝ V α ? 93 of 124

Scaling Geometric argument ◮ Allometrically growing regions: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks L Ω L’ (V’) Earlier theories 2 Ω (V) 2 Geometric argument Blood networks River networks Conclusion References L 1 L’ 1 ◮ 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 94 of 124

Scaling Scaling-at-large Allometry Examples A focus: Metabolism Spherical cows and pancake cows: Measuring exponents History: River networks Earlier theories Geometric argument Blood networks ◮ Question: How does the surface area S cow of our two River networks Conclusion types of cows scale with cow volume V cow ? Insert References question from assignment 3 ( ⊞ ) ◮ Question: For general families of regions, how does surface area S scale with volume V ? Insert question from assignment 3 ( ⊞ ) 95 of 124

Scaling Geometric argument Scaling-at-large Allometry Examples ◮ Best and worst configurations (Banavar et al.) A focus: Metabolism Measuring exponents History: River networks a b Earlier theories Geometric argument Blood networks River networks Conclusion References ◮ Rather obviously: min V net ∝ � distances from source to sinks. 96 of 124

Scaling Minimal network volume: Scaling-at-large Allometry Real supply networks are close to optimal: Examples A focus: Metabolism Measuring exponents History: River networks (a) (b) (c) (d) Earlier theories Geometric argument Blood networks River networks Conclusion References (a) Commuter rail network in the Boston area. The arrow marks Figure 1. 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 [15] : “Shape and efficiency in spatial distribution networks” 97 of 124

Scaling Minimal network volume: Scaling-at-large Allometry Examples A focus: Metabolism Approximate network volume by integral over region: 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 ) Insert question from assignment 3 ( ⊞ ) ∝ ρ V 1 + γ max 98 of 124

Scaling Geometric argument ◮ General result: Scaling-at-large Allometry min V net ∝ ρ V 1 + γ max Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories Geometric argument Blood networks ◮ If scaling is isometric, we have γ max = 1 / d : River networks Conclusion References min V net / iso ∝ ρ V 1 + 1 / d = ρ V ( d + 1 ) / d ◮ 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 99 of 124

Scaling Blood networks Scaling-at-large Allometry Examples A focus: Metabolism ◮ Material costly ⇒ expect lower optimal bound of Measuring exponents V net ∝ ρ V ( d + 1 ) / d to be followed closely. History: River networks Earlier theories Geometric argument ◮ For cardiovascular networks, d = D = 3. Blood networks River networks ◮ Blood volume scales linearly with body volume [40] , Conclusion References V net ∝ V . ◮ Sink density must ∴ decrease as volume increases: ρ ∝ V − 1 / d . ◮ Density of suppliable sinks decreases with organism size. 101 of 124

Scaling Blood networks Scaling-at-large Allometry Examples A focus: Metabolism ◮ Then P , the rate of overall energy use in Ω , can at Measuring exponents History: River networks most scale with volume as Earlier theories Geometric argument Blood networks River networks P ∝ ρ V ∝ ρ M ∝ M ( d − 1 ) / d Conclusion References ◮ For d = 3 dimensional organisms, we have P ∝ M 2 / 3 102 of 124

Scaling Prefactor: Scaling-at-large Stefan-Boltzmann law: ( ⊞ ) Allometry Examples A focus: Metabolism ◮ Measuring exponents d E History: River networks d t = σ ST 4 Earlier theories Geometric argument Blood networks where S is surface and T is temperature. River networks Conclusion ◮ Very rough estimate of prefactor based on scaling of References normal mammalian body temperature and surface 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 . 103 of 124

Scaling River networks Scaling-at-large Allometry Examples ◮ View river networks as collection networks. A focus: Metabolism Measuring exponents ◮ Many sources and one sink. History: River networks Earlier theories ◮ Assume ρ is constant over time: Geometric argument Blood networks River networks V net ∝ ρ V ( d + 1 ) / d = constant × V 3 / 2 Conclusion References ◮ Network volume grows faster than basin ‘volume’ (really area). ◮ It’s all okay: Landscapes are d =2 surfaces living in D =3 dimensions. ◮ Streams can grow not just in width but in depth... 105 of 124

Scaling Hack’s law ◮ Volume of water in river network can be calculated by Scaling-at-large adding up basin areas Allometry Examples ◮ Flows sum in such a way that A focus: Metabolism Measuring exponents History: River networks Earlier theories � V net = a pixel i Geometric argument Blood networks all pixels River networks Conclusion References ◮ Hack’s law again: ℓ ∼ a h ◮ Can argue V net ∝ V 1 + h basin = a 1 + h basin where h is Hack’s exponent. ◮ ∴ minimal volume calculations gives h = 1 / 2 106 of 124

Scaling Real data: Scaling-at-large Allometry Examples A focus: Metabolism ◮ Banavar et al.’s Measuring exponents History: River networks approach [1] is Earlier theories Geometric argument okay because ρ Blood networks River networks really is constant. Conclusion References ◮ The irony: shows optimal basins are isometric ◮ Optimal Hack’s law: ℓ ∼ a h with h = 1 / 2 ◮ (Zzzzz) From Banavar et al. (1999) [1] 107 of 124

Scaling Even better—prefactors match up: Scaling-at-large 20 Amazon Allometry 19 Examples Mississippi A focus: Metabolism Measuring exponents 18 Congo log 10 water volume V [m 3 ] History: River networks Earlier theories 17 Nile Geometric argument Blood networks 16 River networks Conclusion 15 References 14 13 12 11 10 9 8 6 7 8 9 10 11 12 13 log 10 area a [m 2 ] 108 of 124

Scaling Yet more theoretical madness from the Quarterologists: Scaling-at-large Allometry Examples A focus: Metabolism Measuring exponents History: River networks Earlier theories ◮ Banavar et al., 2010, PNAS: Geometric argument Blood networks River networks “A general basis for quarter-power scaling in Conclusion animals.” [2] References ◮ “It has been known for decades that the metabolic rate of animals scales with body mass with an exponent that is almost always < 1, > 2 / 3, and often very close to 3 / 4.” ◮ Cough, cough, cough, hack, wheeze, cough. 109 of 124