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Scaling in Biology and in Society
Definition of scaling: How do attributes of a system change as the system’s size increases?
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How do attributes of organisms change as their body mass increases?
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How do attributes of cities change as their population increases?
Proportionality
Proportionality
y ∝ x means y = c x, where c is a constant.
Proportionality
y ∝ x means y = c x, where c is a constant. For example
y = 2 x
Proportionality
y ∝ x means y = c x, where c is a constant. For example
y = 2 x
or
y = (− ⅓) x
Proportionality
y ∝ x means y = c x, where c is a constant. For example
y = 2 x
or
y = (− ⅓) x
Proportionality is a linear relationship
Scaling Example
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom.
2 × Room Length
x
Room Length
x
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.
2 × Room Length
x
Room Length
x
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.
2 × Room Length
x
Room Length
x
Bed Length
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.
2 × Room Length
x
Room Length
x
Bed Length
2 × Bed Length
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.
2 × Room Length
x
Room Length
x
Bed Length
2 × Bed Length
bed length ∝room length
Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.
2 × Room Length
x
Room Length
x
Bed Length
2 × Bed Length
bed length ∝room length
bed length
room length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
2 × Room Length
x
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The area of a (square) rug you can fit in the room scales quadratically with the length of the room.
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom. rug area ∝(room length)2
Room Length
Scaling Example
x
2 × Room Length
The area of a (square) rug you can fit in the room scales quadratically with the length of the room.
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom. rug area ∝(room length)2
Room Length
Scaling Example
x
2 × Room Length
The area of a (square) rug you can fit in the room scales quadratically with the length of the room.
rug area
room length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The volume of laundry you can pile to the ceiling scales cubically with the length of the room.
(Assume the length, width, and height of the room are approximately equal.)
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The volume of laundry you can pile to the ceiling scales cubically with the length of the room.
(Assume the length, width, and height of the room are approximately equal.)
volume of laundry ∝(room length)3
laundry volume
room length
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The volume of laundry you can pile to the ceiling scales cubically with the length of the room.
(Assume the length, width, and height of the room are approximately equal.)
volume of laundry ∝(room length)3
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The volume of laundry you can pile to the ceiling scales cubically with the length of the room.
(Assume the length, width, and height of the room are approximately equal.)
volume of laundry ∝(room length)3
More generally, many attributes have power-law scaling: attribute ∝(size)α , where α is a constant
x
Suppose you move from a small (square) bedroom to a larger (square) bedroom.
Room Length
Scaling Example
x
2 × Room Length
The volume of laundry you can pile to the ceiling scales cubically with the length of the room.
(Assume the length, width, and height of the room are approximately equal.)
volume of laundry ∝(room length)3
More generally, many attributes have power-law scaling: attribute ∝(size)α , where α is a constant
That is,
attribute =c(size) α , where α is a constant or y = c xα
Power-Law Scaling
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
y = cxα
log y =α log x + logc
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
y = cxα
log y =α log x + logc
y
x
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
36
Straight line on a log-log plot
y = cxα
log y =α log x + logc
y
x
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
37
Straight line on a log-log plot
y = cxα
log y =α log x + logc
y
x
log y
log x
Power-Law Scaling
Two equivalent mathematical expressions for a power law:
38
Straight line on a log-log plot
y = cxα
log y =α log x + logc
y
x
log y
log x
Slope is α
Examples of power law scaling in nature
Gutenberg-Richter law of earthquake magnitudes
Metabolic scaling in animals
K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important? Cambridge, 1984
L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, 912–913, 2010
Scaling crime, income, etc. with city population
Metabolic Scaling in Biology
Metabolic Scaling in Biology
• Metabolic rate: Amount of energy expended by an organism per unit time.
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.
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?
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?
Theories of metabolic scaling
Theories of metabolic scaling
• Early on, some assumptions were made:
Theories of metabolic scaling
• Early on, some assumptions were made – Body is made of cells, in which metabolic reactions take place.
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.
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
Mouse
Radius ∝ r
Mouse
Radius ∝ r
Surface Area ∝ r2
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal
Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal
Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal
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)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal
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)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Note that Surface Area ∝ (Volume)2/3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Note that Surface Area ∝ (Volume)2/3
because Surface Area ∝ r2 = (r3)2/3
∝ (Volume)2/3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Note that Surface Area ∝ (Volume)2/3
because Surface Area ∝ r2 = (r3)2/3
∝ (Volume)2/3
Hypothesis 2: metabolic rate ∝ (body mass)2/3
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Note that Surface Area ∝ (Volume)2/3
because Surface Area ∝ r2 = (r3)2/3
∝ (Volume)2/3
Hypothesis 2: metabolic rate ∝ (body mass)2/3 (“Surface Hypothesis”)
Mouse
Radius ∝ r
Surface Area ∝ r2
Volume ∝ r3
Hamster
Radius ∝ 2 r
Surface Area ∝ 4 r2
Volume ∝ 8 r3
Hippo
Radius ∝ 50 r
Surface Area ∝ 2500 r2
Volume ∝ 125,000 r3
Note that Surface Area ∝ (Volume)2/3
because Surface Area ∝ r2 = (r3)2/3
∝ (Volume)2/3
Hypothesis 2: metabolic rate ∝ (body mass)2/3 (“Surface Hypothesis”)
This was believed for many years!
Actual data:
Actual data: metabolic rate ∝ (body mass)3/4
Actual data: metabolic rate ∝ (body mass)3/4
Hypothesis 3 (“Kleiber’s law): metabolic rate ∝ mass3/4
Actual data: metabolic rate ∝ (body mass)3/4
Hypothesis 3 (“Kleiber’s law): metabolic rate ∝ mass3/4
For sixty years, no explanation
y=x2/3
y=x3/4
y=x3/4
y=x2/3
More “efficient”, in sense that metabolic rate (and thus rate of distribution of nutrients to cells) is larger than surface area would predict.
Other Observed Biological Scaling Laws
Heart rate ∝ body mass-1/4
Blood circulation time ∝ body mass1/4
Life span ∝ body mass1/4
Growth rate ∝ body mass-1/4
Heights of trees ∝ tree mass1/4
Sap circulation time in trees ∝ tree mass1/4
Other Observed Biological Scaling Laws
Heart rate ∝ body mass-1/4
Blood circulation time ∝ body mass1/4
Life span ∝ body mass1/4
Growth rate ∝ body mass-1/4
Heights of trees ∝ tree mass1/4
Sap circulation time in trees ∝ tree mass1/4
No*onof“quarterpowerscaling”
West, Brown, and Enquist’s Theory (1990s)
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.”
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?
Distribution systems
West, Brown, and Enquist’s Theory (1990s)
West, Brown, and Enquist’s Theory (1990s)
• Assumptions about distribution network:
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)
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
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
Because distribution network has fractal branching structure,
Euclidean geometry is the wrong way to view scaling; one should
use fractal geometry instead!
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 mass3/4
West, Brown, and Enquist’s interpretation of their model
• Metabolic rate scales with body mass like surface area scales with volume...
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.
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 four-dimensional. . . Fractal geometry has literally given life an added dimension.”
h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/
Metabolicrate∝ volume2/3(ormass2/3)
h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/
Metabolicrate∝ volume2/3(ormass2/3)
Youare3-dimensional
h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/
Metabolicrate∝ volume2/3(ormass2/3)
Youare3-dimensional
Metabolicrate∝ volume3/4(ormass3/4)
h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/
Metabolicrate∝ volume2/3(ormass2/3)
Youare3-dimensional
Youare4-dimensional
Metabolicrate∝ volume3/4(ormass3/4)
Critiques of their model
Critiques of their model
Critiques of their model
Critiques of their model
Bottom line: Model is interesting and elegant, but both the explanation and the underlying data are controversial.
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.
L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, 912–913, 2010
Do fractal distribution networks explain scaling in cities?
Wrapping Up
Large networks of simple interacting elements,
which, following simple rules, produce emergent,
collective, complex behavior.
What are Complex Systems?
Core Disciplines of the Sciences of Complexity
Dynamics: The study of continually changing structure and behavior of
systems
Information: The study of representation, symbols, and communication
Computation: The study of how systems process information and act on the
results
Evolution / Learning: The study of how systems adapt to constantly
changing environments
Goals of this course:
• To give you a sense of how these topics are integrated in the study of complex systems
• To give you a sense of how idealized models can be used to study these topics
What did we cover?
Let’s review...
Dynamics and Chaos
• Provides a “vocabulary” for describing how complex systems change over time – Fixed points, periodic attractors, chaos, sensitive dependence on initial
conditions
• Shows how complex behavior can arise from iteration of simple rules
• Characterizes complexity in terms of dynamics
• Shows contrast between intrinsic unpredictability and “universal” properties
Fractals
• Provides geometry of real-world patterns
• Shows how complex patterns can arise from iteration of simple rules
• Characterizes complexity in terms of fractal dimension
Information Theory
• Makes analogy between information and physical entropy
• Characterizes complexity in terms of information content
Genetic Algorithms
• Provides idealized models of evolution and adaptation
• Demonstrates how complex behavior/shape can emerge from simple rules (of evolution)
Cellular Automata
• Idealized models of complex systems
• Shows how complex patterns can emerge from iterating simple rules
• Characterizes complexity in terms of “class” of patterns
Models of Self-Organization
• Idealized models of self-organizing behavior
• Attempt to find common principles in terms of dynamics, information, computation, and adaptation
Firefly synchronization Flocking / Schooling Ant Foraging
Ant Task Allocation Immune System Cellular Metabolism, …
Models of Cooperation
• Idealized model of how self-organized cooperation can emerge in social systems
• Demonstrates how idealized models can be used to study complex phenomena
Prisoner’s dilemma El Farol Problem
Networks
• Vocabulary for describing structure and dynamics of real-world networks – small-world, scale-free, degree distribution, clustering,
path-length
• Shows how real-world network structure can be captured by simple models (e.g., preferential attachment)
Scaling
• Gives clues to underlying structure and dynamics of complex systems (e.g., fractal distribution networks)
Goals of the Science of Complexity
• Cross-disciplinary insights into complex systems
• General theory?
√
?
Can we develop a general theory of complex systems?
That is, a mathematical language that unifies dynamics,
information processing, and evolution in complex systems ?
I.e., a “calculus of complexity” ?
Isaac Newton, 1643–1727
infinitesimal
limit
derivative
integral
“He was hampered by the chaos of language
—words still vaguely defined and words not
quite existing. . . . Newton believed he could
marshal a complete science of motion, if only
he could find the appropriate lexicon. . . .”
― James Gleick, Isaac Newton
emergence
self-organization
network
adaptation
Complex Systems, c. 2013
attractor criticality
information computation
bifurcation
nonlinearity
equilibrium
entropy fractal chaos
.
.
.
.
.
.renormalization randomness
scaling
power law
“I do not give a fig for the simplicity on this side of
complexity, but I would give my life for the simplicity on
the other side of complexity.”
― O. W. Holmes (attr.)