branching networks ii horton tokunaga€¦ · branching networks ii horton ⇔ tokunaga reducing...

BranchingNetworks II

Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 1/74

Branching Networks IIComplex Networks, Course 303A, Spring, 2009

Prof. Peter Dodds

Department of Mathematics & StatisticsUniversity of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 2/74

Outline

Horton ⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 3/74

Can Horton and Tokunaga be happy?

Horton and Tokunaga seem different:

I In terms of network achitecture, Horton’s lawsappear to contain less detailed information thanTokunaga’s law.

I Oddly, Horton’s laws have four parameters andTokunaga has two parameters.

I Rn, Ra, R`, and Rs versus T1 and RT . One simpleredundancy: R` = Rs.Insert question from assignment 1 ()

I To make a connection, clearest approach is to startwith Tokunaga’s law...

I Known result: Tokunaga → Horton [18, 19, 20, 9, 2]


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 4/74

Let us make them happy

We need one more ingredient:

Space-fillingness

I A network is space-filling if the average distancebetween adjacent streams is roughly constant.

I Reasonable for river and cardiovascular networksI For river networks:

Drainage density ρdd = inverse of typical distancebetween channels in a landscape.

I In terms of basin characteristics:

ρdd '∑

stream segment lengthsbasin area

=

∑Ωω=1 nωsω

aΩ

http://www.uvm.edu/~pdodds/teaching/courses/2009-01UVM-303/docs/2009-01UVM-303assignment01.pdf


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 5/74

More with the happy-making thing

Start with Tokunaga’s law: Tk = T1Rk−1T

I Start looking for Horton’s stream number law:nω/nω+1 = Rn.

I Estimate nω, the number of streams of order ω interms of other nω′ , ω′ > ω.

I Observe that each stream of order ω terminates byeither:

ω=3

ω=4

ω=3

ω=3

ω=4

ω=4

1. Running into another stream of order ωand generating a stream of order ω + 1...

I 2nω+1 streams of order ω do this

2. Running into and being absorbed by astream of higher order ω′ > ω...

I n′ωTω′−ω streams of order ω do this


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 6/74

More with the happy-making thing

Putting things together:

I

nω = 2nω+1︸︷︷︸generation

+Ω∑

ω′=ω+1

Tω′−ωnω′︸︷︷︸absorption

I Use Tokunaga’s law and manipulate expression tocreate Rn’s.

I Insert question from assignment 1 ()I Solution:

Rn =(2 + RT + T1)±

√(2 + RT + T1)2 − 8RT

2

(The larger value is the one we want.)


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 7/74

Finding other Horton ratios

Connect Tokunaga to Rs

I Now use uniform drainage density ρdd.I Assume side streams are roughly separated by

distance 1/ρdd.I For an order ω stream segment, expected length is

sω ' ρ−1dd

(1 +

ω−1∑k=1

Tk

)

I Substitute in Tokunaga’s law Tk = T1Rk−1T :

sω ' ρ−1dd

(1 + T1

ω−1∑k=1

R k−1T

)∝ R ω

T


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 8/74

Horton and Tokunaga are happy

Altogether then:

I

⇒ sω/sω−1 = RT ⇒ Rs = RT

I Recall R` = Rs so

R` = RT

I And from before:

Rn =(2 + RT + T1) +

√(2 + RT + T1)2 − 8RT

2



Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 9/74


Some observations:I Rn and R` depend on T1 and RT .I Seems that Ra must as well...I Suggests Horton’s laws must contain some

redundancyI We’ll in fact see that Ra = Rn.I Also: Both Tokunaga’s law and Horton’s laws can be

generalized to relationships between non-trivialstatistical distributions. [3, 4]


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 10/74


The other way round

I Note: We can invert the expresssions for Rn and R`

to find Tokunaga’s parameters in terms of Horton’sparameters.

I

RT = R`,

I

T1 = Rn − R` − 2 + 2R`/Rn.

I Suggests we should be able to argue that Horton’slaws imply Tokunaga’s laws (if drainage density isuniform)...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 11/74

Horton and Tokunaga are friends

From Horton to Tokunaga [2]

(Rl)(a)(b)(c)

I Assume Horton’s lawshold for number andlength

I Start with an order ωstream

I Scale up by a factor ofR`, orders increment

I Maintain drainagedensity by adding neworder 1 streams


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 12/74


. . . and in detail:I Must retain same drainage density.I Add an extra (R` − 1) first order streams for each

original tributary.I Since number of first order streams is now given by

Tk+1 we have:

Tk+1 = (R` − 1)

(k∑

i=1

Ti + 1

).

I For large ω, Tokunaga’s law is the solution—let’scheck...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 13/74


Just checking:

I Substitute Tokunaga’s law Ti = T1R i−1T = T1R i−1

`

into

Tk+1 = (R` − 1)

(k∑

i=1

Ti + 1

)I

Tk+1 = (R` − 1)

(k∑

i=1

T1R i−1` + 1

)

= (R` − 1)T1

(R k

` − 1R` − 1

+ 1)

' (R` − 1)T1R k

`

R` − 1= T1Rk

` ... yep.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 14/74

Horton’s laws of area and number:

1 2 3 4 5 6 7 8 9 10 11

0

1

2

3

4

5

6

7

ω

(a)

The Mississippi

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1 2 3 4 5 6 7 8 9 10 11−7

−6

−5

−4

−3

−2

−1

0

ω

(b)

The Mississippi

Ω = 11

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1 2 3 4 5 6 7 8 9 10 1110

−1

100

101

102

103

104

105

106

107

stream order ω

The Nile

nω

aω (sq km)

lω (km)

[sou

rce=

/dat

a11/

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river

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.ps]

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1 2 3 4 5 6 7 8 9 10 1110

−1

100

101

102

103

104

105

106

107

stream order ω

The Nile

nΩ−ω+1aω (sq km)

Ω = 10

[sou

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I In right plots, stream number graph has been flippedvertically.

I Highly suggestive that Rn ≡ Ra...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 15/74

Measuring Horton ratios is tricky:

I How robust are our estimates of ratios?I Rule of thumb: discard data for two smallest and two

largest orders.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 16/74

Mississippi:

ω range Rn Ra R` Rs Ra/Rn[2, 3] 5.27 5.26 2.48 2.30 1.00[2, 5] 4.86 4.96 2.42 2.31 1.02[2, 7] 4.77 4.88 2.40 2.31 1.02[3, 4] 4.72 4.91 2.41 2.34 1.04[3, 6] 4.70 4.83 2.40 2.35 1.03[3, 8] 4.60 4.79 2.38 2.34 1.04[4, 6] 4.69 4.81 2.40 2.36 1.02[4, 8] 4.57 4.77 2.38 2.34 1.05[5, 7] 4.68 4.83 2.36 2.29 1.03[6, 7] 4.63 4.76 2.30 2.16 1.03[7, 8] 4.16 4.67 2.41 2.56 1.12

mean µ 4.69 4.85 2.40 2.33 1.04std dev σ 0.21 0.13 0.04 0.07 0.03

σ/µ 0.045 0.027 0.015 0.031 0.024


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 17/74

Amazon:

ω range Rn Ra R` Rs Ra/Rn[2, 3] 4.78 4.71 2.47 2.08 0.99[2, 5] 4.55 4.58 2.32 2.12 1.01[2, 7] 4.42 4.53 2.24 2.10 1.02[3, 5] 4.45 4.52 2.26 2.14 1.01[3, 7] 4.35 4.49 2.20 2.10 1.03[4, 6] 4.38 4.54 2.22 2.18 1.03[5, 6] 4.38 4.62 2.22 2.21 1.06[6, 7] 4.08 4.27 2.05 1.83 1.05

mean µ 4.42 4.53 2.25 2.10 1.02std dev σ 0.17 0.10 0.10 0.09 0.02

σ/µ 0.038 0.023 0.045 0.042 0.019


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 18/74

Reducing Horton’s laws:

Rough first effort to show Rn ≡ Ra:

I aΩ ∝ sum of all stream lengths in a order Ω basin(assuming uniform drainage density)

I So:

aΩ 'Ω∑

ω=1

nωsω/ρdd

∝Ω∑

ω=1

R Ω−ωn ·

nΩ︷︸︸︷1︸︷︷︸

nω

s1 · R ω−1s︸︷︷︸

sω

=R Ω

nRs

s1

Ω∑ω=1

(Rs

Rn

)ω


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 19/74


Continued ...I

aΩ ∝RΩ

nRs

s1

Ω∑ω=1

(Rs

Rn

)ω

=RΩ

nRs

s1Rs

Rn

1− (Rs/Rn)Ω

1− (Rs/Rn)

∼ RΩ−1n s1

11− (Rs/Rn)

as Ω

I So, aΩ is growing like R Ωn and therefore:

Rn ≡ Ra


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 20/74


Not quite:

I ... But this only a rough argument as Horton’s lawsdo not imply a strict hierarchy

I Need to account for sidebranching.I Insert question from assignment 1 ()



Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 21/74

Equipartitioning:

Intriguing division of area:

I Observe: Combined area of basins of order ωindependent of ω.

I Not obvious: basins of low orders not necessarilycontained in basis on higher orders.

I Story:Rn ≡ Ra ⇒ nωaω = const

I Reason:nω ∝ (Rn)

−ω

aω ∝ (Ra)ω ∝ n−1

ω


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 22/74

Equipartitioning:Some examples:

1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Mississippi basin partitioning

[sou

rce=

/dat

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1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Amazon basin partitioning

[sou

rce=

/dat

a6/d

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azon

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1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Nile basin partitioning

[sou

rce=

/dat

a11/

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s/w

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river

s/de

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HY

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Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 23/74

Scaling laws

The story so far:

I Natural branching networks are hierarchical,self-similar structures

I Hierarchy is mixedI Tokunaga’s law describes detailed architecture:

Tk = T1Rk−1T .

I We have connected Tokunaga’s and Horton’s lawsI Only two Horton laws are independent (Rn = Ra)I Only two parameters are independent:

(T1, RT )⇔ (Rn, Rs)


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 24/74

Scaling laws

A little further...I Ignore stream ordering for the momentI Pick a random location on a branching network p.I Each point p is associated with a basin and a longest

stream lengthI Q: What is probability that the p’s drainage basin has

area a? P(a) ∝ a−τ for large aI Q: What is probability that the longest stream from p

has length `? P(`) ∝ `−γ for large `

I Roughly observed: 1.3 . τ . 1.5 and 1.7 . γ . 2.0


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 25/74

Scaling laws

Probability distributions with power-law decays

I We see them everywhere:I Earthquake magnitudes (Gutenberg-Richter law)I City sizes (Zipf’s law)I Word frequency (Zipf’s law) [21]

I Wealth (maybe not—at least heavy tailed)I Statistical mechanics (phase transitions) [5]

I A big part of the story of complex systemsI Arise from mechanisms: growth, randomness,

optimization, ...I Our task is always to illuminate the mechanism...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 26/74

Scaling laws

Connecting exponents

I We have the detailed picture of branching networks(Tokunaga and Horton)

I Plan: Derive P(a) ∝ a−τ and P(`) ∝ `−γ starting withTokunaga/Horton story [17, 1, 2]

I Let’s work on P(`)...I Our first fudge: assume Horton’s laws hold

throughout a basin of order Ω.I (We know they deviate from strict laws for low ω and

high ω but not too much.)I Next: place stick between teeth. Bite stick. Proceed.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 27/74

Scaling laws

Finding γ:

I Often useful to work with cumulative distributions,especially when dealing with power-law distributions.

I The complementary cumulative distribution turns outto be most useful:

P>(`∗) = P(` > `∗) =

∫ `max

`=`∗

P(`)d`

I

P>(`∗) = 1− P(` < `∗)

I Also known as the exceedance probability.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 28/74

Scaling lawsFinding γ:

I The connection between P(x) and P>(x) when P(x)has a power law tail is simple:

I Given P(`) ∼ `−γ large ` then for large enough `∗

P>(`∗) =

∫ `max

`=`∗

P(`) d`

∼∫ `max

`=`∗

`−γd`

=`−γ+1

−γ + 1

∣∣∣∣`max

`=`∗

∝ `−γ+1∗ for `max `∗


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 29/74

Scaling laws

Finding γ:

I Aim: determine probability of randomly choosing apoint on a network with main stream length > `∗

I Assume some spatial sampling resolution ∆

I Landscape is broken up into grid of ∆×∆ sitesI Approximate P>(`∗) as

P>(`∗) =N>(`∗;∆)

N>(0;∆).

where N>(`∗;∆) is the number of sites with mainstream length > `∗.

I Use Horton’s law of stream segments:sω/sω−1 = Rs...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 30/74

Scaling laws

Finding γ:

I Set `∗ = `ω for some 1 ω Ω.I

P>(`ω) =N>(`ω;∆)

N>(0;∆)'∑Ω

ω′=ω+1 nω′sω′/∆∑Ωω′=1 nω′sω′/∆

I ∆’s cancelI Denominator is aΩρdd, a constant.I So... using Horton’s laws...

P>(`ω) ∝Ω∑

ω′=ω+1

nω′sω′ 'Ω∑

ω′=ω+1

(1·R Ω−ω′n )(s1·R ω′−1

s )


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 31/74

Scaling laws

Finding γ:

I We are here:

P>(`ω) ∝Ω∑

ω′=ω+1

(1 · R Ω−ω′n )(s1 · R ω′−1

s )

I Cleaning up irrelevant constants:

P>(`ω) ∝Ω∑

ω′=ω+1

(Rs

Rn

)ω′

I Change summation order by substitutingω′′ = Ω− ω′.

I Sum is now from ω′′ = 0 to ω′′ = Ω− ω − 1(equivalent to ω′ = Ω down to ω′ = ω + 1)


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 32/74

Scaling laws

Finding γ:

I

P>(`ω) ∝Ω−ω−1∑ω′′=0

(Rs

Rn

)Ω−ω′′

∝Ω−ω−1∑ω′′=0

(Rn

Rs

)ω′′

I Since Rn > Rs and 1 ω Ω,

P>(`ω) ∝(

Rn

Rs

)Ω−ω

∝(

Rn

Rs

)−ω

again using∑n−1

i=0 ai = (a n − 1)/(a− 1)


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 33/74

Scaling laws

Finding γ:

I Nearly there:

P>(`ω) ∝(

Rn

Rs

)−ω

= e−ω ln(Rn/Rs)

I Need to express right hand side in terms of `ω.I Recall that `ω ' ¯1R ω−1

` .I

`ω ∝ R ω` = R ω

s = e ω ln Rs


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 34/74

Scaling lawsFinding γ:

I Therefore:

P>(`ω) ∝ e−ω ln(Rn/Rs) =(

e ω ln Rs)− ln(Rn/Rs)/ ln(Rs)

I

∝ `ω− ln(Rn/Rs)/ ln Rs

I

= `−(ln Rn−ln Rs)/ ln Rsω

I

= `− ln Rn/ ln Rs+1ω

I

= `−γ+1ω


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 35/74

Scaling laws

Finding γ:

I And so we have:

γ = ln Rn/ ln Rs

I Proceeding in a similar fashion, we can show

τ = 2− ln Rs/ ln Rn = 2− 1/γ

Insert question from assignment 1 ()I Such connections between exponents are called

scaling relationsI Let’s connect to one last relationship: Hack’s law


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 36/74

Scaling laws

Hack’s law: [6]

I

` ∝ ah

I Typically observed that 0.5 . h . 0.7.I Use Horton laws to connect h to Horton ratios:

`ω ∝ R ωs and aω ∝ R ω

n

I Observe:

`ω ∝ e ω ln Rs ∝(

e ω ln Rn)ln Rs/ ln Rn

∝ (R ωn )ln Rs/ ln Rn ∝ a ln Rs/ ln Rn

ω ⇒ h = ln Rs/ ln Rn



Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 37/74

Connecting exponentsOnly 3 parameters are independent:e.g., take d , Rn, and Rs

relation: scaling relation/parameter: [2]

` ∼ Ld dTk = T1(RT )k−1 T1 = Rn − Rs − 2 + 2Rs/Rn

RT = Rsnω/nω+1 = Rn Rnaω+1/aω = Ra Ra = Rn¯ω+1/¯

ω = R` R` = Rs` ∼ ah h = log Rs/ log Rna ∼ LD D = d/h

L⊥ ∼ LH H = d/h − 1P(a) ∼ a−τ τ = 2− hP(`) ∼ `−γ γ = 1/h

Λ ∼ aβ β = 1 + hλ ∼ Lϕ ϕ = d


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 38/74

Equipartitioning reexamined:Recall this story:

1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Mississippi basin partitioning

[sou

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1 2 3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Amazon basin partitioning

[sou

rce=

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a6/d

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1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

ω

n ω a

ω /

a Ω

Nile basin partitioning

[sou

rce=

/dat

a11/

dodd

s/w

ork/

river

s/de

ms/

HY

DR

O1K

/afr

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nile

/figu

res/

figeq

uipa

rt_n

ile.p

s]

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Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

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References

Frame 39/74

Equipartitioning

I What aboutP(a) ∼ a−τ ?

I Since τ > 1, suggests no equipartitioning:

aP(a) ∼ a−τ+1 6= const

I P(a) overcounts basins within basins...I while stream ordering separates basins...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

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References

Frame 40/74

Fluctuations

Moving beyond the mean:

I Both Horton’s laws and Tokunaga’s law relateaverage properties, e.g.,

sω/sω−1 = Rs

I Natural generalization to consideration relationshipsbetween probability distributions

I Yields rich and full description of branching networkstructure

I See into the heart of randomness...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

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Frame 41/74

A toy model—Scheidegger’s model

Directed random networks [11, 12]

I

I

P() = P() = 1/2

I Flow is directed downwardsI Useful and interesting test case—more later...


Horton ⇔Tokunaga

Reducing Horton

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Fluctuations

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References

Frame 42/74

Generalizing Horton’s laws

I ¯ω ∝ (R`)

ω ⇒ N(`|ω) = (RnR`)−ωF`(`/Rω

` )

I aω ∝ (Ra)ω ⇒ N(a|ω) = (R2

n)−ωFa(a/Rωn )

0 100 200 300 40010

−4

10−2

100

102

l (km)

N(l

|ω)

Mississippi: length distributions

ω=3 4 5 6

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

lw_c

olla

pse_

mis

pi2.

ps]

[09−

Dec

−19

99 p

eter

dod

ds]

0 1 2 310

−7

10−6

10−5

10−4

10−3

l Rl−ω

Rnω

−Ω

Rlω N

(l |ω

)


ω=3 4 5 6

Rn = 4.69, R

l = 2.38

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

lw_c

olla

pse_

mis

pi2a

.ps]

[09−

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99 p

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I Scaling collapse works well for intermediate ordersI All moments grow exponentially with order


Horton ⇔Tokunaga

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References

Frame 43/74


I How well does overall basin fit internal pattern?

0 1 2 3 4

x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−7

l RlΩ−ω (m)

Rl−

Ω+

ω P

(l R

lΩ−

ω )

Mississippi

ω=4 ω=3 actual l<l>

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

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gure

s/fig

lw_b

low

nup.

ps]

[10−

Dec

−19

99 p

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I Actual length = 4920 km(at 1 km res)

I Predicted Mean length= 11100 km

I Predicted Std dev =5600 km

I Actual length/Meanlength = 44 %

I Okay.


Horton ⇔Tokunaga

Reducing Horton

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Fluctuations

Models

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References

Frame 44/74


Comparison of predicted versus measured main streamlengths for large scale river networks (in 103 km):

basin: `Ω¯Ω σ` `/¯

Ω σ`/¯Ω

Mississippi 4.92 11.10 5.60 0.44 0.51Amazon 5.75 9.18 6.85 0.63 0.75Nile 6.49 2.66 2.20 2.44 0.83Congo 5.07 10.13 5.75 0.50 0.57Kansas 1.07 2.37 1.74 0.45 0.73

a aΩ σa a/aΩ σa/aΩ

Mississippi 2.74 7.55 5.58 0.36 0.74Amazon 5.40 9.07 8.04 0.60 0.89Nile 3.08 0.96 0.79 3.19 0.82Congo 3.70 10.09 8.28 0.37 0.82Kansas 0.14 0.49 0.42 0.28 0.86


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 45/74

Combining stream segments distributions:

I Stream segmentssum to give mainstream lengths

I

`ω =

µ=ω∑µ=1

sµ

I P(`ω) is aconvolution ofdistributions forthe sω


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

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References

Frame 46/74


I Sum of variables `ω =∑µ=ω

µ=1 sµ leads to convolutionof distributions:

N(`|ω) = N(s|1) ∗ N(s|2) ∗ · · · ∗ N(s|ω)

0 1 2 3−7

−6

−5

−4

−3

l (s)ω R

l (s)−ω

log 10

Rnω

−Ω

Rl (

s)ω

P(l (

s) ω, ω

) (b)

Mississippi: stream segments

Rn = 4.69, R

l = 2.33

[sou

rce=

/dat

a6/d

odds

/wor

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ers/

dem

s/m

issi

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_col

laps

e_m

ispi

2a.p

s]

[07−

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−20

00 p

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ds]

N(s|ω) =1

Rωn Rω

`

F (s/Rω` )

F (x) = e−x/ξ

Mississippi: ξ ' 900 m.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

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Models

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References

Frame 47/74


I Next level up: Main stream length distributions mustcombine to give overall distribution for stream length

101

102

10−4

10−2

100

102

l (km)

N(l

|ω)


ω=3 4 5 6 3−6

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

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s/fig

lw_p

ower

law

sum

_mis

pi.p

s]

[22−

Mar

−20

00 p

eter

dod

ds]

I P(`) ∼ `−γ

I Another round ofconvolutions [3]

I Interesting...


Horton ⇔Tokunaga

Reducing Horton

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References

Frame 48/74


Number and areadistributions for theScheidegger modelP(n1,6) versus P(a 6).

0 1 2 3 4

x 104

0

0.2

0.4

0.6

0.8

1

1.2x 10

−4

0 1 2 3

x 104

0

1

2

3

4

x 10−4


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

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References

Frame 49/74

Generalizing Tokunaga’s law

Scheidegger:

0 100 200 300−5

−4

−3

−2

−1

(a)

Tµ,ν

log 10

P(T

µ,ν )

0 0.1 0.2 0.3 0.4 0.5 0.6−3

−2

−1

0

1(b)

Tµ,ν (Rl (s))−µ

log 10

(Rl (

s) )

µ P(T

µ,ν )

I Observe exponential distributions for Tµ,ν

I Scaling collapse works using Rs


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

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References

Frame 50/74


Mississippi:

0 20 40 600

0.5

1

1.5

2

2.5

(a)

Tµ,ν

log 10

P(T

µ,ν )

0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

Tµ,ν (Rl (s))ν

log 10

(Rl (

s) )

−ν P

(Tµ,

ν )

(b)

I Same data collapse for Mississippi...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

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References

Frame 51/74


SoP(Tµ,ν) = (Rs)

µ−ν−1Pt

[Tµ,ν/(Rs)

µ−ν−1]

wherePt(z) =

1ξt

e−z/ξt .

P(sµ)⇔ P(Tµ,ν)

I Exponentials arise from randomness.I Look at joint probability P(sµ, Tµ,ν).


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 52/74


Network architecture:

I Inter-tributarylengthsexponentiallydistributed

I Leads to randomspatial distributionof streamsegments

1 2


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 53/74


I Follow streams segments down stream from theirbeginning

I Probability (or rate) of an order µ stream segmentterminating is constant:

pµ ' 1/(Rs)µ−1ξs

I Probability decays exponentially with stream orderI Inter-tributary lengths exponentially distributedI ⇒ random spatial distribution of stream segments


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 54/74


I Joint distribution for generalized version ofTokunaga’s law:

P(sµ, Tµ,ν) = pµ

(sµ − 1Tµ,ν

)pTµ,ν

ν (1− pν − pµ)sµ−Tµ,ν−1

whereI pν = probability of absorbing an order ν side streamI pµ = probability of an order µ stream terminating

I Approximation: depends on distance units of sµ

I In each unit of distance along stream, there is onechance of a side stream entering or the streamterminating.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 55/74


I Now deal with thing:

P(sµ, Tµ,ν) = pµ

(sµ − 1Tµ,ν

)pTµ,ν

ν (1− pν − pµ)sµ−Tµ,ν−1

I Set (x , y) = (sµ, Tµ,ν) and q = 1− pν − pµ,approximate liberally.

I ObtainP(x , y) = Nx−1/2 [F (y/x)]x

where

F (v) =

(1− v

q

)−(1−v)(vp

)−v

.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 56/74


I Checking form of P(sµ, Tµ,ν) works:

Scheidegger:

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a)

v = Tµ,ν / lµ (s)

[F(v

)]l µ (

s)

0.05 0.1 0.15−1.5

−1

−0.5

0

0.5

1

1.5(b)

v = Tµ,ν / lµ (s)

P(v

| l µ (

s))


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 57/74



Scheidegger:

0 0.1 0.2 0.3−1.5

−1

−0.5

0

0.5

1

1.5(a)

Tµ,ν / lµ (s)

log 10

P(T

µ,ν /

l µ (s) )

0 10 20 30 40 50−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5 (b)

lµ (s) / Tµ,ν

log 10

P(l µ (

s) /

T µ,ν )


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 58/74



Scheidegger:

0 0.05 0.1 0.15−0.5

0

0.5

1

1.5

2 (a)

Tµ,ν / lµ (s)

log 10

P(T

µ,ν /

l µ (s) )

−0.2 −0.1 0 0.1 0.2−1.5

−1

−0.5

0

0.5

1 (b)

[Tµ,ν / lµ (s) − ρν] (R

l (s) )µ/2−ν/2

log 10

(Rl (

s) )

−µ/

2+ν/

2 P(T

µ,ν /

l µ (s) )


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 59/74



Mississippi:

0 0.15 0.3 0.45 0.6

0

0.5

1

1.5

(a)

Tµ,ν / lµ (s)

log 10

P(T

µ,ν /

l µ (s)

)

−0.5 −0.25 0 0.25 0.5−0.8

−0.4

0

0.4

0.8(b)

[Tµ,ν / lµ (s) − ρν](R

l (s))ν

log 10

(Rl (

s))−

ν P(T

µ,ν /

l µ (s) )


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

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References

Frame 60/74

Models

Random subnetworks on a Bethe lattice [13]

I Dominant theoretical conceptfor several decades.

I Bethe lattices are fun andtractable.

I Led to idea of “Statisticalinevitability” of river networkstatistics [7]

I But Bethe latticesunconnected with surfaces.

I In fact, Bethe lattices 'infinite dimensional spaces(oops).

I So let’s move on...


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

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References

Frame 61/74

Scheidegger’s model

Directed random networks [11, 12]

I

I

P() = P() = 1/2

I Functional form of all scaling laws exhibited butexponents differ from real world [15, 16, 14]


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 62/74

A toy model—Scheidegger’s model

Random walk basins:I Boundaries of basins are random walks

n

x

area a


Horton ⇔Tokunaga

Reducing Horton

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Fluctuations

Models

Nutshell

References

Frame 63/74


n

2

6 6

8 8 8 8

9 9Increasing partition of N=64

x


Horton ⇔Tokunaga

Reducing Horton

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Models

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References

Frame 64/74


Prob for first return of a random walk in (1+1)dimensions (from CSYS/MATH 300):

I

P(n) ∼ 12√

πn−3/2.

and so P(`) ∝ `−3/2.I Typical area for a walk of length n is ∝ n3/2:

` ∝ a 2/3.

I Find τ = 4/3, h = 2/3, γ = 3/2, d = 1.I Note τ = 2− h and γ = 1/h.I Rn and R` have not been derived analytically.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

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Nutshell

References

Frame 65/74

Optimal channel networks

Rodríguez-Iturbe, Rinaldo, et al. [10]

I Landscapes h(~x) evolve such that energy dissipationε is minimized, where

ε ∝∫

d~r (flux)× (force) ∼∑

i

ai∇hi ∼∑

i

aγi

I Landscapes obtained numerically give exponentsnear that of real networks.

I But: numerical method used matters.I And: Maritan et al. find basic universality classes are

that of Scheidegger, self-similar, and a third kind ofrandom network [8]


Horton ⇔Tokunaga

Reducing Horton

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Fluctuations

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References

Frame 66/74

Theoretical networks

Summary of universality classes:

network h dNon-convergent flow 1 1

Directed random 2/3 1Undirected random 5/8 5/4

Self-similar 1/2 1OCN’s (I) 1/2 1OCN’s (II) 2/3 1OCN’s (III) 3/5 1Real rivers 0.5–0.7 1.0–1.2

h ⇒ ` ∝ ah (Hack’s law).d ⇒ ` ∝ Ld

‖ (stream self-affinity).


Horton ⇔Tokunaga

Reducing Horton

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References

Frame 67/74

Nutshell

Branching networks II Key Points:

I Horton’s laws and Tokunaga law all fit together.I nb. for 2-d networks, these laws are ‘planform’ laws

and ignore slope.I Abundant scaling relations can be derived.I Can take Rn, R`, and d as three independent

parameters necessary to describe all 2-d branchingnetworks.

I For scaling laws, only h = ln R`/ ln Rn and d areneeded.

I Laws can be extended nicely to laws of distributions.I Numerous models of branching network evolution

exist: nothing rock solid yet.


Horton ⇔Tokunaga

Reducing Horton

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References

Frame 68/74

References I

H. de Vries, T. Becker, and B. Eckhardt.Power law distribution of discharge in ideal networks.Water Resources Research, 30(12):3541–3543,December 1994.

P. S. Dodds and D. H. Rothman.Unified view of scaling laws for river networks.Physical Review E, 59(5):4865–4877, 1999. pdf ()

P. S. Dodds and D. H. Rothman.Geometry of river networks. II. Distributions ofcomponent size and number.Physical Review E, 63(1):016116, 2001. pdf ()

P. S. Dodds and D. H. Rothman.Geometry of river networks. III. Characterization ofcomponent connectivity.Physical Review E, 63(1):016117, 2001. pdf ()

http://www.uvm.edu/~pdodds/research/papers/others/1999/dodds1999a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2001/dodds2001b.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2001/dodds2001c.pdf


Horton ⇔Tokunaga

Reducing Horton

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References

Frame 69/74

References II

N. Goldenfeld.Lectures on Phase Transitions and theRenormalization Group, volume 85 of Frontiers inPhysics.Addison-Wesley, Reading, Massachusetts, 1992.

J. T. Hack.Studies of longitudinal stream profiles in Virginia andMaryland.United States Geological Survey Professional Paper,294-B:45–97, 1957.

J. W. Kirchner.Statistical inevitability of Horton’s laws and theapparent randomness of stream channel networks.Geology, 21:591–594, July 1993.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 70/74

References III

A. Maritan, F. Colaiori, A. Flammini, M. Cieplak, andJ. R. Banavar.Universality classes of optimal channel networks.Science, 272:984–986, 1996. pdf ()

S. D. Peckham.New results for self-similar trees with applications toriver networks.Water Resources Research, 31(4):1023–1029, April1995.

I. Rodríguez-Iturbe and A. Rinaldo.Fractal River Basins: Chance and Self-Organization.Cambridge University Press, Cambrigde, UK, 1997.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

Nutshell

References

Frame 71/74

References IV

A. E. Scheidegger.A stochastic model for drainage patterns into anintramontane trench.Bull. Int. Assoc. Sci. Hydrol., 12(1):15–20, 1967.

A. E. Scheidegger.Theoretical Geomorphology.Springer-Verlag, New York, third edition, 1991.

R. L. Shreve.Infinite topologically random channel networks.Journal of Geology, 75:178–186, 1967.

H. Takayasu.Steady-state distribution of generalized aggregationsystem with injection.Physcial Review Letters, 63(23):2563–2565,December 1989.


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

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References

Frame 72/74

References V

H. Takayasu, I. Nishikawa, and H. Tasaki.Power-law mass distribution of aggregation systemswith injection.Physical Review A, 37(8):3110–3117, April 1988.

M. Takayasu and H. Takayasu.Apparent independency of an aggregation systemwith injection.Physical Review A, 39(8):4345–4347, April 1989.

D. G. Tarboton, R. L. Bras, and I. Rodríguez-Iturbe.Comment on “On the fractal dimension of streamnetworks” by Paolo La Barbera and Renzo Rosso.Water Resources Research, 26(9):2243–4,September 1990.

http://www.uvm.edu/~pdodds/research/papers/others/1996/maritan1996b.pdf


Horton ⇔Tokunaga

Reducing Horton

Scaling relations

Fluctuations

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Nutshell

References

Frame 73/74

References VI

E. Tokunaga.The composition of drainage network in ToyohiraRiver Basin and the valuation of Horton’s first law.Geophysical Bulletin of Hokkaido University, 15:1–19,1966.

E. Tokunaga.Consideration on the composition of drainagenetworks and their evolution.Geographical Reports of Tokyo MetropolitanUniversity, 13:G1–27, 1978.

E. Tokunaga.Ordering of divide segments and law of dividesegment numbers.Transactions of the Japanese GeomorphologicalUnion, 5(2):71–77, 1984.


Horton ⇔Tokunaga

Reducing Horton

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Fluctuations

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References

Frame 74/74

References VII

G. K. Zipf.Human Behaviour and the Principle of Least-Effort.Addison-Wesley, Cambridge, MA, 1949.

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