Download Mechanisms for Generating Power-Law Size Distributions I Principles of Complex Systems

Power-Law
Mechanisms I
Mechanisms for Generating
Power-Law Size Distributions I
Random Walks
The First Return Problem
Examples
Principles of Complex Systems
CSYS/MATH 300, Fall, 2011
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
Prof. Peter Dodds
References
Department of Mathematics & Statistics | Center for Complex Systems |
Vermont Advanced Computing Center | University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
1 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
2 of 56
Mechanisms
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
A powerful story in the rise of complexity:
I
References
structure arises out of randomness.
3 of 56
Mechanisms
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
A powerful story in the rise of complexity:
I
References
structure arises out of randomness.
3 of 56
Mechanisms
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
A powerful story in the rise of complexity:
I
structure arises out of randomness.
I
Exhibit A: Random walks... ()
References
3 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
The essential random walk:
I
One spatial dimension.
I
Time and space are discrete
I
Random walker (e.g., a drunk) starts at origin x = 0.
I
Step at time t is t :
+1 with probability 1/2
t =
−1 with probability 1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
4 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
The essential random walk:
I
One spatial dimension.
I
Time and space are discrete
I
Random walker (e.g., a drunk) starts at origin x = 0.
I
Step at time t is t :
+1 with probability 1/2
t =
−1 with probability 1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
4 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
The essential random walk:
I
One spatial dimension.
I
Time and space are discrete
I
Random walker (e.g., a drunk) starts at origin x = 0.
I
Step at time t is t :
+1 with probability 1/2
t =
−1 with probability 1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
4 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
The essential random walk:
I
One spatial dimension.
I
Time and space are discrete
I
Random walker (e.g., a drunk) starts at origin x = 0.
I
Step at time t is t :
+1 with probability 1/2
t =
−1 with probability 1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
4 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Displacement after t steps:
Basics
xt =
t
X
Holtsmark’s Distribution
PLIPLO
i
References
i=1
Expected displacement:
* t +
t
X
X
hxt i =
i =
hi i = 0
i=1
i=1
5 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Displacement after t steps:
Basics
xt =
t
X
Holtsmark’s Distribution
PLIPLO
i
References
i=1
Expected displacement:
* t +
t
X
X
hxt i =
i =
hi i = 0
i=1
i=1
5 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Displacement after t steps:
Basics
xt =
t
X
Holtsmark’s Distribution
PLIPLO
i
References
i=1
Expected displacement:
* t +
t
X
X
hi i = 0
hxt i =
i =
i=1
i=1
5 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Displacement after t steps:
Basics
xt =
t
X
Holtsmark’s Distribution
PLIPLO
i
References
i=1
Expected displacement:
* t +
t
X
X
hi i = 0
hxt i =
i =
i=1
i=1
5 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Variances sum: ()∗
Basics
Var(xt ) = Var
t
X
!
i
Holtsmark’s Distribution
PLIPLO
References
i=1
=
t
X
i=1
Var (i ) =
t
X
1=t
i=1
∗
Sum rule = a good reason for using the variance to measure
spread; only works for independent distributions.
6 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Variances sum: ()∗
Basics
Var(xt ) = Var
t
X
!
i
Holtsmark’s Distribution
PLIPLO
References
i=1
=
t
X
i=1
Var (i ) =
t
X
1=t
i=1
∗
Sum rule = a good reason for using the variance to measure
spread; only works for independent distributions.
6 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Variances sum: ()∗
Basics
Var(xt ) = Var
t
X
!
i
Holtsmark’s Distribution
PLIPLO
References
i=1
=
t
X
i=1
Var (i ) =
t
X
1=t
i=1
∗
Sum rule = a good reason for using the variance to measure
spread; only works for independent distributions.
6 of 56
Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
So typical displacement from the origin scales as
PLIPLO
References
σ = t 1/2
⇒ A non-trivial power-law arises out of
additive aggregation or accumulation.
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Power-Law
Mechanisms I
Random walks
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
So typical displacement from the origin scales as
PLIPLO
References
σ = t 1/2
⇒ A non-trivial power-law arises out of
additive aggregation or accumulation.
7 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is...
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is...
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is...
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is...
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is...
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is... 0.
Holtsmark’s Distribution
PLIPLO
References
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random walks are weirder than you might think...
Variable
transformation
Basics
For example:
I
ξr ,t = the probability that by time step t, a random
walk has crossed the origin r times.
I
Think of a coin flip game with ten thousand tosses.
I
If you are behind early on, what are the chances you
will make a comeback?
I
The most likely number of lead changes is... 0.
Holtsmark’s Distribution
PLIPLO
References
See Feller, [2] Intro to Probability Theory, Volume I
8 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
In fact:
PLIPLO
ξ0,t > ξ1,t > ξ2,t > · · ·
References
Even crazier:
9 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
In fact:
PLIPLO
ξ0,t > ξ1,t > ξ2,t > · · ·
References
Even crazier:
The expected time between tied scores = ∞!
9 of 56
Power-Law
Mechanisms I
Random walks—some examples
Random Walks
The First Return Problem
Examples
50
Variable
transformation
x
0
−50
Basics
Holtsmark’s Distribution
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
t
PLIPLO
References
50
x
0
−50
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
6000
7000
8000
9000
10000
t
200
x
100
0
−100
0
1000
2000
3000
4000
5000
t
10 of 56
Power-Law
Mechanisms I
Random walks—some examples
Random Walks
The First Return Problem
Examples
50
Variable
transformation
x
0
−50
Basics
Holtsmark’s Distribution
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
t
PLIPLO
References
200
x
100
0
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
6000
7000
8000
9000
10000
t
200
x
100
0
−100
0
1000
2000
3000
4000
5000
t
11 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
The problem of first return:
I
What is the probability that a random walker in one
dimension returns to the origin for the first time after t
steps?
I
Will our drunkard always return to the origin?
I
What about higher dimensions?
Holtsmark’s Distribution
PLIPLO
References
12 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
The problem of first return:
I
What is the probability that a random walker in one
dimension returns to the origin for the first time after t
steps?
I
Will our drunkard always return to the origin?
I
What about higher dimensions?
Holtsmark’s Distribution
PLIPLO
References
12 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
The problem of first return:
I
What is the probability that a random walker in one
dimension returns to the origin for the first time after t
steps?
I
Will our drunkard always return to the origin?
I
What about higher dimensions?
Holtsmark’s Distribution
PLIPLO
References
12 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
The problem of first return:
I
What is the probability that a random walker in one
dimension returns to the origin for the first time after t
steps?
I
Will our drunkard always return to the origin?
I
What about higher dimensions?
Holtsmark’s Distribution
PLIPLO
References
12 of 56
First returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Reasons for caring:
Basics
Holtsmark’s Distribution
PLIPLO
1. We will find a power-law size distribution with an
interesting exponent
References
2. Some physical structures may result from random
walks
3. We’ll start to see how different scalings relate to
each other
13 of 56
First returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Reasons for caring:
Basics
Holtsmark’s Distribution
PLIPLO
1. We will find a power-law size distribution with an
interesting exponent
References
2. Some physical structures may result from random
walks
3. We’ll start to see how different scalings relate to
each other
13 of 56
First returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Reasons for caring:
Basics
Holtsmark’s Distribution
PLIPLO
1. We will find a power-law size distribution with an
interesting exponent
References
2. Some physical structures may result from random
walks
3. We’ll start to see how different scalings relate to
each other
13 of 56
First returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Reasons for caring:
Basics
Holtsmark’s Distribution
PLIPLO
1. We will find a power-law size distribution with an
interesting exponent
References
2. Some physical structures may result from random
walks
3. We’ll start to see how different scalings relate to
each other
13 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
14 of 56
Power-Law
Mechanisms I
Random Walks
Random Walks
The First Return Problem
Examples
Variable
transformation
50
0
x
Basics
Holtsmark’s Distribution
−50
PLIPLO
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
6000
7000
8000
9000
10000
References
t
200
x
100
0
−100
0
1000
2000
3000
4000
5000
t
15 of 56
Power-Law
Mechanisms I
Random Walks
Random Walks
The First Return Problem
Examples
Variable
transformation
50
0
x
Basics
Holtsmark’s Distribution
−50
PLIPLO
−100
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
6000
7000
8000
9000
10000
References
t
200
x
100
0
−100
0
1000
2000
3000
4000
5000
t
Again: expected time between ties = ∞...
Let’s find out why... [2]
15 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
4
Basics
Holtsmark’s Distribution
PLIPLO
x
2
References
0
−2
−4
0
5
10
15
20
t
16 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
For random walks in 1-d:
I
Return can only happen when t = 2n.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
17 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
For random walks in 1-d:
I
Return can only happen when t = 2n.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Call Pfirst return (2n) = Pfr (2n) probability of first return
at t = 2n.
References
17 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
For random walks in 1-d:
I
Return can only happen when t = 2n.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Call Pfirst return (2n) = Pfr (2n) probability of first return
at t = 2n.
I
Assume drunkard first lurches to x = 1.
References
17 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
For random walks in 1-d:
I
Return can only happen when t = 2n.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Call Pfirst return (2n) = Pfr (2n) probability of first return
at t = 2n.
I
Assume drunkard first lurches to x = 1.
I
The problem
References
Pfr (2n) = 2Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x2n = 0)
17 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
x
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
1
References
0
0
2
4
6
8
10
12
14
16
t
I
A useful restatement: Pfr (2n) =
2 · 12 Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x1 = x2n−1 = 1)
I
Want walks that can return many times to x = 1.
I
(The 12 accounts for stepping to 2 instead of 0 at
t = 2n.)
18 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
x
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
1
References
0
0
2
4
6
8
10
12
14
16
t
I
A useful restatement: Pfr (2n) =
2 · 12 Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x1 = x2n−1 = 1)
I
Want walks that can return many times to x = 1.
I
(The 12 accounts for stepping to 2 instead of 0 at
t = 2n.)
18 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
x
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
1
References
0
0
2
4
6
8
10
12
14
16
t
I
A useful restatement: Pfr (2n) =
2 · 12 Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x1 = x2n−1 = 1)
I
Want walks that can return many times to x = 1.
I
(The 12 accounts for stepping to 2 instead of 0 at
t = 2n.)
18 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
x
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
1
References
0
0
2
4
6
8
10
12
14
16
t
I
A useful restatement: Pfr (2n) =
2 · 12 Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x1 = x2n−1 = 1)
I
Want walks that can return many times to x = 1.
I
(The 12 accounts for stepping to 2 instead of 0 at
t = 2n.)
18 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
x
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
1
References
0
0
2
4
6
8
10
12
14
16
t
I
A useful restatement: Pfr (2n) =
2 · 12 Pr (xt ≥ 1, t = 1, . . . , 2n − 1, and x1 = x2n−1 = 1)
I
Want walks that can return many times to x = 1.
I
(The 12 accounts for stepping to 2 instead of 0 at
t = 2n.)
18 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Counting problem (combinatorics/statistical
mechanics)
I
Use a method of images
I
Define N(i, j, t) as the # of possible walks between
x = i and x = j taking t steps.
I
Consider all paths starting at x = 1 and ending at
x = 1 after t = 2n − 2 steps.
I
Subtract how many hit x = 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
19 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Counting problem (combinatorics/statistical
mechanics)
I
Use a method of images
I
Define N(i, j, t) as the # of possible walks between
x = i and x = j taking t steps.
I
Consider all paths starting at x = 1 and ending at
x = 1 after t = 2n − 2 steps.
I
Subtract how many hit x = 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
19 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Counting problem (combinatorics/statistical
mechanics)
I
Use a method of images
I
Define N(i, j, t) as the # of possible walks between
x = i and x = j taking t steps.
I
Consider all paths starting at x = 1 and ending at
x = 1 after t = 2n − 2 steps.
I
Subtract how many hit x = 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
19 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Counting problem (combinatorics/statistical
mechanics)
I
Use a method of images
I
Define N(i, j, t) as the # of possible walks between
x = i and x = j taking t steps.
I
Consider all paths starting at x = 1 and ending at
x = 1 after t = 2n − 2 steps.
I
Subtract how many hit x = 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
19 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Counting problem (combinatorics/statistical
mechanics)
I
Use a method of images
I
Define N(i, j, t) as the # of possible walks between
x = i and x = j taking t steps.
I
Consider all paths starting at x = 1 and ending at
x = 1 after t = 2n − 2 steps.
I
Subtract how many hit x = 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
19 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Key observation:
# of t-step paths starting and ending at x = 1
and hitting x = 0 at least once
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
20 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Key observation:
# of t-step paths starting and ending at x = 1
and hitting x = 0 at least once
= # of t-step paths starting at x = −1 and ending at x = 1
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
20 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Key observation:
# of t-step paths starting and ending at x = 1
and hitting x = 0 at least once
= # of t-step paths starting at x = −1 and ending at x = 1
= N(−1, 1, t)
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
20 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Key observation:
# of t-step paths starting and ending at x = 1
and hitting x = 0 at least once
= # of t-step paths starting at x = −1 and ending at x = 1
= N(−1, 1, t)
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
So Nfirst return (2n) = N(1, 1, 2n − 2) − N(−1, 1, 2n − 2)
20 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Key observation:
# of t-step paths starting and ending at x = 1
and hitting x = 0 at least once
= # of t-step paths starting at x = −1 and ending at x = 1
= N(−1, 1, t)
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
So Nfirst return (2n) = N(1, 1, 2n − 2) − N(−1, 1, 2n − 2)
See this 1-1 correspondence visually...
20 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
References
x
1
0
−1
−2
−3
−4
0
2
4
6
8
10
12
14
16
t
21 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
For any path starting at x = 1 that hits 0,
there is a unique matching path starting at x = −1.
I
Matching path first mirrors and then tracks.
References
22 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
For any path starting at x = 1 that hits 0,
there is a unique matching path starting at x = −1.
I
Matching path first mirrors and then tracks.
References
22 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
4
The First Return Problem
Examples
3
Variable
transformation
Basics
2
Holtsmark’s Distribution
PLIPLO
References
x
1
0
−1
−2
−3
−4
0
2
4
6
8
10
12
14
16
t
23 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
I
Next problem: what is N(i, j, t)?
I
# positive steps + # negative steps = t.
I
Random walk must displace by j − i after t steps.
I
# positive steps - # negative steps = j − i.
I
# positive steps = (t + j − i)/2.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
N(i, j, t) =
t
t
=
# positive steps
(t + j − i)/2
24 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
We now have
Holtsmark’s Distribution
PLIPLO
References
Nfirst return (2n) = N(1, 1, 2n − 2) − N(−1, 1, 2n − 2)
where
N(i, j, t) =
t
(t + j − i)/2
25 of 56
Power-Law
Mechanisms I
First Returns
Insert question from assignment 4 ()
Random Walks
The First Return Problem
Examples
Find Nfirst return (2n) ∼
22n−3/2
√
.
2πn3/2
Variable
transformation
Basics
Holtsmark’s Distribution
I
Normalized Number of Paths gives Probability
I
Total number of possible paths = 22n
I
Pfirst return (2n) =
'
PLIPLO
References
1
Nfirst return (2n)
22n
1 22n−3/2
√
22n 2πn3/2
1
= √ (2n)−3/2
2π
26 of 56
Power-Law
Mechanisms I
First Returns
Insert question from assignment 4 ()
Random Walks
The First Return Problem
Examples
Find Nfirst return (2n) ∼
22n−3/2
√
.
2πn3/2
Variable
transformation
Basics
Holtsmark’s Distribution
I
Normalized Number of Paths gives Probability
I
Total number of possible paths = 22n
I
Pfirst return (2n) =
'
PLIPLO
References
1
Nfirst return (2n)
22n
1 22n−3/2
√
22n 2πn3/2
1
= √ (2n)−3/2
2π
26 of 56
Power-Law
Mechanisms I
First Returns
Insert question from assignment 4 ()
Random Walks
The First Return Problem
Examples
Find Nfirst return (2n) ∼
22n−3/2
√
.
2πn3/2
Variable
transformation
Basics
Holtsmark’s Distribution
I
Normalized Number of Paths gives Probability
I
Total number of possible paths = 22n
I
Pfirst return (2n) =
'
PLIPLO
References
1
Nfirst return (2n)
22n
1 22n−3/2
√
22n 2πn3/2
1
= √ (2n)−3/2
2π
26 of 56
Power-Law
Mechanisms I
First Returns
Insert question from assignment 4 ()
Random Walks
The First Return Problem
Examples
Find Nfirst return (2n) ∼
22n−3/2
√
.
2πn3/2
Variable
transformation
Basics
Holtsmark’s Distribution
I
Normalized Number of Paths gives Probability
I
Total number of possible paths = 22n
I
Pfirst return (2n) =
'
PLIPLO
References
1
Nfirst return (2n)
22n
1 22n−3/2
√
22n 2πn3/2
1
= √ (2n)−3/2
2π
26 of 56
Power-Law
Mechanisms I
First Returns
Insert question from assignment 4 ()
Random Walks
The First Return Problem
Examples
Find Nfirst return (2n) ∼
22n−3/2
√
.
2πn3/2
Variable
transformation
Basics
Holtsmark’s Distribution
I
Normalized Number of Paths gives Probability
I
Total number of possible paths = 22n
I
Pfirst return (2n) =
'
PLIPLO
References
1
Nfirst return (2n)
22n
1 22n−3/2
√
22n 2πn3/2
1
= √ (2n)−3/2
2π
26 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Same scaling holds for continuous space/time walks.
Basics
Holtsmark’s Distribution
PLIPLO
I
P(t) ∝ t −3/2 , γ = 3/2
I
P(t) is normalizable
I
Recurrence: Random walker always returns to origin
I
Moral: Repeated gambling against an infinitely
wealthy opponent must lead to ruin.
References
27 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Same scaling holds for continuous space/time walks.
Basics
Holtsmark’s Distribution
PLIPLO
I
P(t) ∝ t −3/2 , γ = 3/2
I
P(t) is normalizable
I
Recurrence: Random walker always returns to origin
I
Moral: Repeated gambling against an infinitely
wealthy opponent must lead to ruin.
References
27 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Same scaling holds for continuous space/time walks.
Basics
Holtsmark’s Distribution
PLIPLO
I
P(t) ∝ t −3/2 , γ = 3/2
I
P(t) is normalizable
I
Recurrence: Random walker always returns to origin
I
Moral: Repeated gambling against an infinitely
wealthy opponent must lead to ruin.
References
27 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Same scaling holds for continuous space/time walks.
Basics
Holtsmark’s Distribution
PLIPLO
I
P(t) ∝ t −3/2 , γ = 3/2
I
P(t) is normalizable
I
Recurrence: Random walker always returns to origin
I
Moral: Repeated gambling against an infinitely
wealthy opponent must lead to ruin.
References
27 of 56
Power-Law
Mechanisms I
First Returns
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Same scaling holds for continuous space/time walks.
Basics
Holtsmark’s Distribution
PLIPLO
I
P(t) ∝ t −3/2 , γ = 3/2
I
P(t) is normalizable
I
Recurrence: Random walker always returns to origin
I
Moral: Repeated gambling against an infinitely
wealthy opponent must lead to ruin.
References
27 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Higher dimensions:
I
Walker in d = 2 dimensions must also return
I
Walker may not return in d ≥ 3 dimensions
I
For d = 1, γ = 3/2 → hti = ∞
I
Even though walker must return, expect a long wait...
Holtsmark’s Distribution
PLIPLO
References
28 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Higher dimensions:
I
Walker in d = 2 dimensions must also return
I
Walker may not return in d ≥ 3 dimensions
I
For d = 1, γ = 3/2 → hti = ∞
I
Even though walker must return, expect a long wait...
Holtsmark’s Distribution
PLIPLO
References
28 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Higher dimensions:
I
Walker in d = 2 dimensions must also return
I
Walker may not return in d ≥ 3 dimensions
I
For d = 1, γ = 3/2 → hti = ∞
I
Even though walker must return, expect a long wait...
Holtsmark’s Distribution
PLIPLO
References
28 of 56
First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Higher dimensions:
I
Walker in d = 2 dimensions must also return
I
Walker may not return in d ≥ 3 dimensions
I
For d = 1, γ = 3/2 → hti = ∞
I
Even though walker must return, expect a long wait...
Holtsmark’s Distribution
PLIPLO
References
28 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
On finite spaces:
Variable
transformation
Basics
Holtsmark’s Distribution
I
In any finite volume, a random walker will visit every
site with equal probability
I
Random walking ≡ Diffusion
I
Call this probability the Invariant Density of a
dynamical system
I
Non-trivial Invariant Densities arise in chaotic
systems.
PLIPLO
References
29 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
On finite spaces:
Variable
transformation
Basics
Holtsmark’s Distribution
I
In any finite volume, a random walker will visit every
site with equal probability
I
Random walking ≡ Diffusion
I
Call this probability the Invariant Density of a
dynamical system
I
Non-trivial Invariant Densities arise in chaotic
systems.
PLIPLO
References
29 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
On finite spaces:
Variable
transformation
Basics
Holtsmark’s Distribution
I
In any finite volume, a random walker will visit every
site with equal probability
I
Random walking ≡ Diffusion
I
Call this probability the Invariant Density of a
dynamical system
I
Non-trivial Invariant Densities arise in chaotic
systems.
PLIPLO
References
29 of 56
Random walks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
On finite spaces:
Variable
transformation
Basics
Holtsmark’s Distribution
I
In any finite volume, a random walker will visit every
site with equal probability
I
Random walking ≡ Diffusion
I
Call this probability the Invariant Density of a
dynamical system
I
Non-trivial Invariant Densities arise in chaotic
systems.
PLIPLO
References
29 of 56
Random walks on
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
On networks:
I
On networks, a random walker visits each node with
frequency ∝ node degree
I
Equal probability still present:
walkers traverse edges with equal frequency.
PLIPLO
References
30 of 56
Random walks on
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
On networks:
I
On networks, a random walker visits each node with
frequency ∝ node degree
I
Equal probability still present:
walkers traverse edges with equal frequency.
PLIPLO
References
30 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
31 of 56
Scheidegger Networks [4, 1]
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
Triangular lattice
I
‘Flow’ is southeast or southwest with equal
probability.
32 of 56
Scheidegger Networks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Creates basins with random walk boundaries
I
Observe Subtracting one random walk from another
gives random walk with increments

 +1 with probability 1/4
0 with probability 1/2
t =

−1 with probability 1/4
Variable
transformation
Basics
Holtsmark’s Distribution
I
PLIPLO
References
Basin length ` distribution: P(`) ∝ `−3/2
33 of 56
Scheidegger Networks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Creates basins with random walk boundaries
I
Observe Subtracting one random walk from another
gives random walk with increments

 +1 with probability 1/4
0 with probability 1/2
t =

−1 with probability 1/4
Variable
transformation
Basics
Holtsmark’s Distribution
I
PLIPLO
References
Basin length ` distribution: P(`) ∝ `−3/2
33 of 56
Scheidegger Networks
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
Creates basins with random walk boundaries
I
Observe Subtracting one random walk from another
gives random walk with increments

 +1 with probability 1/4
0 with probability 1/2
t =

−1 with probability 1/4
Variable
transformation
Basics
Holtsmark’s Distribution
I
PLIPLO
References
Basin length ` distribution: P(`) ∝ `−3/2
33 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
I
For a basin of length `, width ∝
I
Basin area a ∝ ` · `1/2 = `3/2
a2/3
I
Invert: ` ∝
I
d` ∝ d(a2/3 ) = 2/3a−1/3 da
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−3/2 d`
∝ (a2/3 )−3/2 a−1/3 da
= a−4/3 da
= a−τ da
`1/2
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
34 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Both basin area and length obey power law
distributions
I
Observed for real river networks
I
Typically: 1.3 < τ < 1.5 and 1.5 < γ < 2
I
Smaller basins more allometric (h > 1/2)
I
Larger basins more isometric (h = 1/2)
Holtsmark’s Distribution
PLIPLO
References
35 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Both basin area and length obey power law
distributions
I
Observed for real river networks
I
Typically: 1.3 < τ < 1.5 and 1.5 < γ < 2
I
Smaller basins more allometric (h > 1/2)
I
Larger basins more isometric (h = 1/2)
Holtsmark’s Distribution
PLIPLO
References
35 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Both basin area and length obey power law
distributions
I
Observed for real river networks
I
Typically: 1.3 < τ < 1.5 and 1.5 < γ < 2
I
Smaller basins more allometric (h > 1/2)
I
Larger basins more isometric (h = 1/2)
Holtsmark’s Distribution
PLIPLO
References
35 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Both basin area and length obey power law
distributions
I
Observed for real river networks
I
Typically: 1.3 < τ < 1.5 and 1.5 < γ < 2
I
Smaller basins more allometric (h > 1/2)
I
Larger basins more isometric (h = 1/2)
Holtsmark’s Distribution
PLIPLO
References
35 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
I
I
Generalize relationship between area and length
PLIPLO
References
Hack’s law [3] :
` ∝ ah
where 0.5 . h . 0.7
I
Redo calc with γ, τ , and h.
36 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
I
I
Generalize relationship between area and length
PLIPLO
References
Hack’s law [3] :
` ∝ ah
where 0.5 . h . 0.7
I
Redo calc with γ, τ , and h.
36 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
I
I
Generalize relationship between area and length
PLIPLO
References
Hack’s law [3] :
` ∝ ah
where 0.5 . h . 0.7
I
Redo calc with γ, τ , and h.
36 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
I
The First Return Problem
Given
Examples
` ∝ ah , P(a) ∝ a−τ , and P(`) ∝ `−γ
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
d(ah )
hah−1 da
I
d` ∝
I
Pr (basin area = a)da
= Pr (basin length = `)d`
∝ `−γ d`
∝ (ah )−γ ah−1 da
= a−(1+h (γ−1)) da
=
References
I
τ = 1 + h(γ − 1)
37 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
With more detailed description of network structure,
τ = 1 + h(γ − 1) simplifies:
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
τ =2−h
PLIPLO
References
γ = 1/h
38 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
With more detailed description of network structure,
τ = 1 + h(γ − 1) simplifies:
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
τ =2−h
PLIPLO
References
γ = 1/h
I
Only one exponent is independent
38 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
With more detailed description of network structure,
τ = 1 + h(γ − 1) simplifies:
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
τ =2−h
PLIPLO
References
γ = 1/h
I
Only one exponent is independent
I
Simplify system description
38 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
With more detailed description of network structure,
τ = 1 + h(γ − 1) simplifies:
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
τ =2−h
PLIPLO
References
γ = 1/h
I
Only one exponent is independent
I
Simplify system description
I
Expect scaling relations where power laws are found
38 of 56
Connections between Exponents
Power-Law
Mechanisms I
Random Walks
The First Return Problem
With more detailed description of network structure,
τ = 1 + h(γ − 1) simplifies:
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
τ =2−h
PLIPLO
References
γ = 1/h
I
Only one exponent is independent
I
Simplify system description
I
Expect scaling relations where power laws are found
I
Characterize universality class with independent
exponents
38 of 56
Other First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Failure
I
A very simple model of failure/death:
I
xt = entity’s ‘health’ at time t
I
x0 could be > 0.
I
Entity fails when x hits 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Streams
I
Dispersion of suspended sediments in streams.
I
Long times for clearing.
39 of 56
Other First Returns
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Failure
I
A very simple model of failure/death:
I
xt = entity’s ‘health’ at time t
I
x0 could be > 0.
I
Entity fails when x hits 0.
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Streams
I
Dispersion of suspended sediments in streams.
I
Long times for clearing.
39 of 56
More than randomness
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Can generalize to Fractional Random Walks
I
Levy flights, Fractional Brownian Motion
I
In 1-d,
Basics
Holtsmark’s Distribution
PLIPLO
References
σ ∼tα
I
Extensive memory of path now matters...
40 of 56
More than randomness
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Can generalize to Fractional Random Walks
I
Levy flights, Fractional Brownian Motion
I
In 1-d,
Basics
Holtsmark’s Distribution
PLIPLO
References
σ ∼tα
I
Extensive memory of path now matters...
40 of 56
More than randomness
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Can generalize to Fractional Random Walks
I
Levy flights, Fractional Brownian Motion
I
In 1-d,
Basics
Holtsmark’s Distribution
PLIPLO
References
σ ∼tα
I
Extensive memory of path now matters...
40 of 56
More than randomness
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Can generalize to Fractional Random Walks
I
Levy flights, Fractional Brownian Motion
I
In 1-d,
Basics
Holtsmark’s Distribution
PLIPLO
References
σ ∼tα
α > 1/2 — superdiffusive
α < 1/2 — subdiffusive
I
Extensive memory of path now matters...
40 of 56
More than randomness
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Can generalize to Fractional Random Walks
I
Levy flights, Fractional Brownian Motion
I
In 1-d,
Basics
Holtsmark’s Distribution
PLIPLO
References
σ ∼tα
α > 1/2 — superdiffusive
α < 1/2 — subdiffusive
I
Extensive memory of path now matters...
40 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
41 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
Understand power laws as arising from
References
1. elementary distributions (e.g., exponentials)
42 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
Understand power laws as arising from
References
1. elementary distributions (e.g., exponentials)
2. variables connected by power relationships
42 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Random variable X with known distribution Px
I
Second random variable Y with y = f (x).
I
I
Basics
Holtsmark’s Distribution
PLIPLO
References
Py (y )dy = Px (x)dx
=
P
−1 (y ))
y |f (x)=y Px (f
dy Figure...
|f 0 (f −1 (y ))|
Often easier to do by
hand...
43 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Random variable X with known distribution Px
I
Second random variable Y with y = f (x).
I
I
Basics
Holtsmark’s Distribution
PLIPLO
References
Py (y )dy = Px (x)dx
=
P
−1 (y ))
y |f (x)=y Px (f
dy Figure...
|f 0 (f −1 (y ))|
Often easier to do by
hand...
43 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Random variable X with known distribution Px
I
Second random variable Y with y = f (x).
I
I
Basics
Holtsmark’s Distribution
PLIPLO
References
Py (y )dy = Px (x)dx
=
P
−1 (y ))
y |f (x)=y Px (f
dy Figure...
|f 0 (f −1 (y ))|
Often easier to do by
hand...
43 of 56
Variable Transformation
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
I
Random variable X with known distribution Px
I
Second random variable Y with y = f (x).
I
I
Basics
Holtsmark’s Distribution
PLIPLO
References
Py (y )dy = Px (x)dx
=
P
−1 (y ))
y |f (x)=y Px (f
dy Figure...
|f 0 (f −1 (y ))|
Often easier to do by
hand...
43 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
= c(−α)x −α−1 dx
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
= c(−α)x −α−1 dx
invert: dx =
−1 α+1
x
dy
cα
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
= c(−α)x −α−1 dx
−1 α+1
x
dy
cα
−1 y −(α+1)/α
dy
dx =
cα c
invert: dx =
44 of 56
Power-Law
Mechanisms I
General Example
Assume relationship between x and y is 1-1.
I
I
Power-law relationship between variables:
y = cx −α , α > 0
Look at y large and x small
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
Holtsmark’s Distribution
dy = d cx
−α
PLIPLO
References
= c(−α)x −α−1 dx
−1 α+1
x
dy
cα
−1 y −(α+1)/α
dy
dx =
cα c
invert: dx =
dx =
−c 1/α −1−1/α
y
dy
α
44 of 56
General Example
Power-Law
Mechanisms I
Now make transformation:
Random Walks
Py (y )dy = Px (x)dx
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
I
If Px (x) → non-zero constant as x → 0 then
Py (y ) ∝ y −1−1/α as y → ∞.
I
If Px (x) → x β as x → 0 then
Py (y ) ∝ y −1−1/α−β/α as y → ∞.
45 of 56
Power-Law
Mechanisms I
General Example
Now make transformation:
Random Walks
The First Return Problem
Py (y )dy = Px (x)dx
Examples
Variable
transformation
(x)
dx
}|
{
z
}|
{z
y −1/α c 1/α
Py (y )dy = Px
y −1−1/α dy
c
α
I
Basics
Holtsmark’s Distribution
PLIPLO
References
If Px (x) → non-zero constant as x → 0 then
Py (y ) ∝ y −1−1/α as y → ∞.
I
If Px (x) → x β as x → 0 then
Py (y ) ∝ y −1−1/α−β/α as y → ∞.
45 of 56
Power-Law
Mechanisms I
General Example
Now make transformation:
Random Walks
The First Return Problem
Py (y )dy = Px (x)dx
Examples
Variable
transformation
(x)
dx
}|
{
z
}|
{z
y −1/α c 1/α
Py (y )dy = Px
y −1−1/α dy
c
α
I
Basics
Holtsmark’s Distribution
PLIPLO
References
If Px (x) → non-zero constant as x → 0 then
Py (y ) ∝ y −1−1/α as y → ∞.
I
If Px (x) → x β as x → 0 then
Py (y ) ∝ y −1−1/α−β/α as y → ∞.
45 of 56
Power-Law
Mechanisms I
General Example
Now make transformation:
Random Walks
The First Return Problem
Py (y )dy = Px (x)dx
Examples
Variable
transformation
(x)
dx
}|
{
z
}|
{z
y −1/α c 1/α
Py (y )dy = Px
y −1−1/α dy
c
α
I
Basics
Holtsmark’s Distribution
PLIPLO
References
If Px (x) → non-zero constant as x → 0 then
Py (y ) ∝ y −1−1/α as y → ∞.
I
If Px (x) → x β as x → 0 then
Py (y ) ∝ y −1−1/α−β/α as y → ∞.
45 of 56
Power-Law
Mechanisms I
Example
Random Walks
The First Return Problem
Examples
Variable
transformation
Exponential distribution
Given Px (x) =
1 −x/λ
λe
and y =
Basics
Holtsmark’s Distribution
cx −α ,
PLIPLO
then
References
P(y ) ∝ y −1−1/α + O y −1−2/α
46 of 56
Power-Law
Mechanisms I
Example
Random Walks
The First Return Problem
Examples
Variable
transformation
Exponential distribution
Given Px (x) =
1 −x/λ
λe
and y =
Basics
Holtsmark’s Distribution
cx −α ,
PLIPLO
then
References
P(y ) ∝ y −1−1/α + O y −1−2/α
I
Exponentials arise from randomness...
46 of 56
Power-Law
Mechanisms I
Example
Random Walks
The First Return Problem
Examples
Variable
transformation
Exponential distribution
Given Px (x) =
1 −x/λ
λe
and y =
Basics
Holtsmark’s Distribution
cx −α ,
PLIPLO
then
References
P(y ) ∝ y −1−1/α + O y −1−2/α
I
Exponentials arise from randomness...
I
More later when we cover robustness.
46 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
47 of 56
Gravity
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Select a random point in the
universe ~x
I
(possible all of space-time)
I
Measure the force of gravity
F (~x )
I
Observe that PF (F ) ∼ F −5/2 .
References
48 of 56
Gravity
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Select a random point in the
universe ~x
I
(possible all of space-time)
I
Measure the force of gravity
F (~x )
I
Observe that PF (F ) ∼ F −5/2 .
References
48 of 56
Gravity
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Select a random point in the
universe ~x
I
(possible all of space-time)
I
Measure the force of gravity
F (~x )
I
Observe that PF (F ) ∼ F −5/2 .
References
48 of 56
Gravity
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
Select a random point in the
universe ~x
I
(possible all of space-time)
I
Measure the force of gravity
F (~x )
I
Observe that PF (F ) ∼ F −5/2 .
References
48 of 56
Power-Law
Mechanisms I
Ingredients [5]
Matter is concentrated in stars:
I
F is distributed unevenly
I
Probability of being a distance r from a single star at
~x = ~0:
Pr (r )dr ∝ r 2 dr
I
Assume stars are distributed randomly in space
(oops?)
Assume only one star has significant effect at ~x .
I
Law of gravity:
I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
F ∝ r −2
I
invert:
r ∝ F −1/2
49 of 56
Power-Law
Mechanisms I
Ingredients [5]
Matter is concentrated in stars:
I
F is distributed unevenly
I
Probability of being a distance r from a single star at
~x = ~0:
Pr (r )dr ∝ r 2 dr
I
Assume stars are distributed randomly in space
(oops?)
Assume only one star has significant effect at ~x .
I
Law of gravity:
I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
F ∝ r −2
I
invert:
r ∝ F −1/2
49 of 56
Power-Law
Mechanisms I
Ingredients [5]
Matter is concentrated in stars:
I
F is distributed unevenly
I
Probability of being a distance r from a single star at
~x = ~0:
Pr (r )dr ∝ r 2 dr
I
Assume stars are distributed randomly in space
(oops?)
Assume only one star has significant effect at ~x .
I
Law of gravity:
I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
F ∝ r −2
I
invert:
r ∝ F −1/2
49 of 56
Power-Law
Mechanisms I
Ingredients [5]
Matter is concentrated in stars:
I
F is distributed unevenly
I
Probability of being a distance r from a single star at
~x = ~0:
Pr (r )dr ∝ r 2 dr
I
Assume stars are distributed randomly in space
(oops?)
Assume only one star has significant effect at ~x .
I
Law of gravity:
I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
F ∝ r −2
I
invert:
r ∝ F −1/2
49 of 56
Power-Law
Mechanisms I
Ingredients [5]
Matter is concentrated in stars:
I
F is distributed unevenly
I
Probability of being a distance r from a single star at
~x = ~0:
Pr (r )dr ∝ r 2 dr
I
Assume stars are distributed randomly in space
(oops?)
Assume only one star has significant effect at ~x .
I
Law of gravity:
I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
F ∝ r −2
I
invert:
r ∝ F −1/2
49 of 56
Power-Law
Mechanisms I
Transformation
Random Walks
The First Return Problem
Examples
I
dF ∝ d(r −2 )
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ r −3 dr
I
References
invert:
dr ∝ r 3 dF
I
∝ F −3/2 dF
50 of 56
Power-Law
Mechanisms I
Transformation
Random Walks
The First Return Problem
Examples
I
dF ∝ d(r −2 )
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ r −3 dr
I
References
invert:
dr ∝ r 3 dF
I
∝ F −3/2 dF
50 of 56
Power-Law
Mechanisms I
Transformation
Random Walks
The First Return Problem
Examples
I
dF ∝ d(r −2 )
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ r −3 dr
I
References
invert:
dr ∝ r 3 dF
I
∝ F −3/2 dF
50 of 56
Power-Law
Mechanisms I
Transformation
Random Walks
The First Return Problem
Examples
I
dF ∝ d(r −2 )
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ r −3 dr
I
References
invert:
dr ∝ r 3 dF
I
∝ F −3/2 dF
50 of 56
Power-Law
Mechanisms I
Transformation
Using r ∝ F −1/2 , dr ∝ F −3/2 dF and Pr (r ) ∝ r 2
Random Walks
The First Return Problem
Examples
I
PF (F )dF = Pr (r )dr
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ Pr (F −1/2 )F −3/2 dF
I
References
2
∝ F −1/2 F −3/2 dF
I
= F −1−3/2 dF
I
= F −5/2 dF
51 of 56
Power-Law
Mechanisms I
Transformation
Using r ∝ F −1/2 , dr ∝ F −3/2 dF and Pr (r ) ∝ r 2
Random Walks
The First Return Problem
Examples
I
PF (F )dF = Pr (r )dr
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ Pr (F −1/2 )F −3/2 dF
I
References
2
∝ F −1/2 F −3/2 dF
I
= F −1−3/2 dF
I
= F −5/2 dF
51 of 56
Power-Law
Mechanisms I
Transformation
Using r ∝ F −1/2 , dr ∝ F −3/2 dF and Pr (r ) ∝ r 2
Random Walks
The First Return Problem
Examples
I
PF (F )dF = Pr (r )dr
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ Pr (F −1/2 )F −3/2 dF
I
References
2
∝ F −1/2 F −3/2 dF
I
= F −1−3/2 dF
I
= F −5/2 dF
51 of 56
Power-Law
Mechanisms I
Transformation
Using r ∝ F −1/2 , dr ∝ F −3/2 dF and Pr (r ) ∝ r 2
Random Walks
The First Return Problem
Examples
I
PF (F )dF = Pr (r )dr
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ Pr (F −1/2 )F −3/2 dF
I
References
2
∝ F −1/2 F −3/2 dF
I
= F −1−3/2 dF
I
= F −5/2 dF
51 of 56
Power-Law
Mechanisms I
Transformation
Using r ∝ F −1/2 , dr ∝ F −3/2 dF and Pr (r ) ∝ r 2
Random Walks
The First Return Problem
Examples
I
PF (F )dF = Pr (r )dr
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
∝ Pr (F −1/2 )F −3/2 dF
I
References
2
∝ F −1/2 F −3/2 dF
I
= F −1−3/2 dF
I
= F −5/2 dF
51 of 56
Power-Law
Mechanisms I
Gravity
Random Walks
The First Return Problem
Examples
PF (F ) = F −5/2 dF
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
References
γ = 5/2
I
Mean is finite
I
Variance = ∞
I
A wild distribution
I
Random sampling of space usually safe
but can end badly...
52 of 56
Power-Law
Mechanisms I
Gravity
Random Walks
The First Return Problem
Examples
PF (F ) = F −5/2 dF
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
References
γ = 5/2
I
Mean is finite
I
Variance = ∞
I
A wild distribution
I
Random sampling of space usually safe
but can end badly...
52 of 56
Power-Law
Mechanisms I
Gravity
Random Walks
The First Return Problem
Examples
PF (F ) = F −5/2 dF
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
References
γ = 5/2
I
Mean is finite
I
Variance = ∞
I
A wild distribution
I
Random sampling of space usually safe
but can end badly...
52 of 56
Power-Law
Mechanisms I
Gravity
Random Walks
The First Return Problem
Examples
PF (F ) = F −5/2 dF
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
References
γ = 5/2
I
Mean is finite
I
Variance = ∞
I
A wild distribution
I
Random sampling of space usually safe
but can end badly...
52 of 56
Power-Law
Mechanisms I
Gravity
Random Walks
The First Return Problem
Examples
PF (F ) = F −5/2 dF
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
I
References
γ = 5/2
I
Mean is finite
I
Variance = ∞
I
A wild distribution
I
Random sampling of space usually safe
but can end badly...
52 of 56
Outline
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
Variable transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
53 of 56
Caution!
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
PLIPLO = Power law in, power law out
I
Explain a power law as resulting from another
unexplained power law.
I
Yet another homunculus argument ()...
I
Don’t do this!!! (slap, slap)
I
We need mechanisms!
Holtsmark’s Distribution
PLIPLO
References
54 of 56
Caution!
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
PLIPLO = Power law in, power law out
I
Explain a power law as resulting from another
unexplained power law.
I
Yet another homunculus argument ()...
I
Don’t do this!!! (slap, slap)
I
We need mechanisms!
Holtsmark’s Distribution
PLIPLO
References
54 of 56
Caution!
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
PLIPLO = Power law in, power law out
I
Explain a power law as resulting from another
unexplained power law.
I
Yet another homunculus argument ()...
I
Don’t do this!!! (slap, slap)
I
We need mechanisms!
Holtsmark’s Distribution
PLIPLO
References
54 of 56
Caution!
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
Basics
I
PLIPLO = Power law in, power law out
I
Explain a power law as resulting from another
unexplained power law.
I
Yet another homunculus argument ()...
I
Don’t do this!!! (slap, slap)
I
We need mechanisms!
Holtsmark’s Distribution
PLIPLO
References
54 of 56
References I
Power-Law
Mechanisms I
Random Walks
[1] P. S. Dodds and D. H. Rothman.
Scaling, universality, and geomorphology.
Annu. Rev. Earth Planet. Sci., 28:571–610, 2000.
pdf ()
[2] W. Feller.
An Introduction to Probability Theory and Its
Applications, volume I.
John Wiley & Sons, New York, third edition, 1968.
The First Return Problem
Examples
Variable
transformation
Basics
Holtsmark’s Distribution
PLIPLO
References
[3] J. T. Hack.
Studies of longitudinal stream profiles in Virginia and
Maryland.
United States Geological Survey Professional Paper,
294-B:45–97, 1957. pdf ()
55 of 56
References II
Power-Law
Mechanisms I
Random Walks
The First Return Problem
Examples
Variable
transformation
[4] A. E. Scheidegger.
The algebra of stream-order numbers.
United States Geological Survey Professional Paper,
525-B:B187–B189, 1967. pdf ()
Basics
Holtsmark’s Distribution
PLIPLO
References
[5] D. Sornette.
Critical Phenomena in Natural Sciences.
Springer-Verlag, Berlin, 2nd edition, 2003.
56 of 56