Scaling in Complex Systems. Copyright by Melanie Mitchell

Transcription

1 Scaling in Complex Systems 1

2 Definition of scaling: How do attributes of a system change as the system s size increases? 2

3 How do attributes of organisms change as their body mass increases? 3

4 aprilphoto0811/aprilphoto / smallcity-in-italy-is-located-in-a-valley-among-hills-thetop-view.jpg How do attributes of cities change as their population increases? 4

5 Power-Law Scaling Two equivalent mathematical expressions for a power law: y = cx α y x log y = α log x + logc Straight line on a log-log plot log y Slope is α log x 5

6 Examples of power law scaling in nature Gutenberg-Richter law of earthquake magnitudes 6

7 Rank-frequency scaling: City populations 7

8 Rank-frequency scaling: Income distribution 8

9 Rank-frequency scaling: Word frequency in English (Zipf s law) Frequency of word w = 1/rank of word w A plot of word frequency of single words (unigrams) versus rank r extracted from the one million words of the Brown s English dictionary. ( 9

10 Metabolic scaling in animals K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important? Cambridge,

11 Scaling crime, income, etc. with city population L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, ,

12 Controversy: Can Network Structure and Dynamics Explain Scaling in Biology and Other Disciplines? 12

13 Metabolic Scaling in Biology Metabolic rate: Amount of energy expended by an organism per unit time. Can be measured as the amount of heat emitted by the organism per unit time. It has been known for a long time that metabolic rate is a function of body mass, but how, exactly, does metabolic rate scale with body mass? 13

14 Theories of metabolic scaling Early on, some assumptions were made: Body is made of cells, in which metabolic reactions take place. Can approximate body mass by a sphere of cells with radius r. r 14

15 Mouse Radius r Surface Area r 2 Volume r 3 Hamster Radius 2 r Surface Area 4 r 2 Volume 8 r 3 Hypothesis 1: metabolic rate body mass (where body mass volume) BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal Hippo Radius 50 r Surface Area 2500 r 2 Volume 125,000 r 3 15

16 Mouse Radius r Surface Area r 2 Volume r 3 Hamster Radius 2 r Surface Area 4 r 2 Volume 8 r 3 125,000 times the heat of a mouse radiating over an area only 2500 times the surface area of a mouse Hypothesis 1: metabolic rate body mass (where body mass volume) BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal Hippo Radius 50 r Surface Area 2500 r 2 Volume 125,000 r 3 16

17 Mouse Radius r Surface Area r 2 Volume r 3 Hamster Radius 2 r Surface Area 4 r 2 Volume 8 r 3 Note that Surface Area (Volume) 2/3 because Surface Area r 2 = (r 3 ) 2/3 (Volume) 2/3 This was believed for many years! Hypothesis 2: metabolic rate (body mass) 2/3 Hippo Radius 50 r ( Surface Hypothesis ) Surface Area 2500 r 2 Volume 125,000 r 3 17

18 Actual data: metabolic rate (body mass) 3/4 For sixty years, no explanation Hypothesis 3 ( Kleiber s law): metabolic rate mass 3/4 18

19 More efficient, in sense that metabolic rate (and thus rate of distribution of nutrients to cells) is larger than surface area would predict. y = x 3/4 y = x 2/3 19

20 Other Observed Biological Scaling Laws Heart rate body mass -1/4 Blood circulation time body mass 1/4 Life span body mass 1/4 Growth rate body mass -1/4 No7on of quarter power scaling Heights of trees tree mass 1/4 Sap circulation time in trees tree mass 1/4 20

21 West, Brown, and Enquist s Theory (1990s) 21

22 West, Brown, and Enquist s Theory (1990s) General idea: metabolic scaling rates (and other biological rates) are limited not by surface area but by rates at which energy and materials can be distributed between surfaces where they are exchanged and the tissues where they are used. How are energy and materials distributed? 22

23 Distribution systems 23

24 West, Brown, and Enquist s Theory (1990s) Assumptions about distribution network: branches to reach all parts of three-dimensional organism (i.e., needs to be as space-filling as possible) has terminal units (e.g., capillaries) that do not vary with size among organisms evolved to minimize total energy required to distribute resources 24

25 Because distribution network has fractal branching structure, Euclidean geometry is the wrong way to view scaling; one should use fractal geometry instead! With detailed mathematical model using three assumptions, they derive metabolic rate body mass 3/4 25

26 West, Brown, and Enquist s interpretation of their model Metabolic rate scales with body mass like surface area scales with volume... but in four dimensions. Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were fourdimensional... Fractal geometry has literally given life an added dimension. 26

27 Metabolic rate volume 2/3 (or mass 2/3 ) hap:// he- daily- insight the- big- hand- view- of- the- human- body/ You are 3- dimensional Metabolic rate volume 3/4 (or mass 3/4 ) You are 4- dimensional 27

28 Critiques of their model Bottom line: Model is interesting and elegant, but both the explanation and the underlying data are controversial. Also, note that there have been many updated versions of their model since their original paper. 28

29 L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, , 2010 Do fractal distribution networks explain scaling in cities? 29