outline branching networks introduction

BranchingNetworks

Introduction

River NetworksDefinitions

Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 1/121

Branching NetworksComplex Networks, Course 295A, Spring, 2008

Prof. Peter Dodds

Department of Mathematics & StatisticsUniversity of Vermont

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 2/121

Outline

Introduction

River NetworksDefinitionsAllometryLawsStream OrderingHorton’s LawsTokunaga’s LawHorton ⇔ TokunagaReducing HortonScaling relationsFluctuationsModels

References

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 3/121

Introduction

Branching networks are useful things:

I Fundamental to material supply and collectionI Supply: From one source to many sinks in 2- or 3-d.I Collection: From many sources to one sink in 2- or

3-d.I Typically observe hierarchical, recursive self-similar

structure

Examples:

I River networks (our focus)I Cardiovascular networksI PlantsI Evolutionary treesI Organizations (only in theory...)

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 4/121

Branching networks are everywhere...

http://hydrosheds.cr.usgs.gov/ ()

http://hydrosheds.cr.usgs.gov/

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 5/121

Branching networks are everywhere...

http://en.wikipedia.org/wiki/Image:Applebox.JPG ()

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 7/121

Geomorphological networks

DefinitionsI Drainage basin for a point p is the complete region of

land from which overland flow drains through p.I Definition most sensible for a point in a stream.I Recursive structure: Basins contain basins and so

on.I In principle, a drainage basin is defined at every

point on a landscape.I On flat hillslopes, drainage basins are effectively

linear.I We treat subsurface and surface flow as following the

gradient of the surface.I Okay for large-scale networks...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 8/121

Basic basin quantities: a, l , L‖, L⊥:

aL?0

L? Lk = La0 ll0Lk0

I a = drainagebasin area

I ` = length oflongest (main)stream (whichmay be fractal)

I L = L‖ =longitudinal lengthof basin

I L = L⊥ = width ofbasin

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 10/121

Allometry

Isometry: dimensions scale linearly with each other.

Allometry: dimensions scale nonlinearly.

http://en.wikipedia.org/wiki/Image:Applebox.JPG

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 11/121

Basin allometry

aL?0

L? Lk = La0 ll0Lk0

Allometricrelationships:

I

` ∝ ah

I

` ∝ Ld

I Combine above:

a ∝ Ld/h ≡ LD

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 12/121

‘Laws’I Hack’s law (1957) [6]:

` ∝ ah

reportedly 0.5 < h < 0.7

I Scaling of main stream length with basin size:

` ∝ Ld‖

reportedly 1.0 < d < 1.1

I Basin allometry:

L‖ ∝ ah/d ≡ a1/D

D < 2 → basins elongate.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 14/121

There are a few more ‘laws’: [2]

Relation: Name or description:

Tk = T1(RT )k Tokunaga’s law` ∼ Ld self-affinity of single channels

nω/nω+1 = Rn Horton’s law of stream numbers¯ω+1/¯

ω = R` Horton’s law of main stream lengthsaω+1/aω = Ra Horton’s law of basin areassω+1/sω = Rs Horton’s law of stream segment lengths

L⊥ ∼ LH scaling of basin widthsP(a) ∼ a−τ probability of basin areasP(`) ∼ `−γ probability of stream lengths

` ∼ ah Hack’s lawa ∼ LD scaling of basin areasΛ ∼ aβ Langbein’s lawλ ∼ Lϕ variation of Langbein’s law

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 15/121

Reported parameter values: [2]

Parameter: Real networks:

Rn 3.0–5.0Ra 3.0–6.0

R` = RT 1.5–3.0T1 1.0–1.5d 1.1± 0.01D 1.8± 0.1h 0.50–0.70τ 1.43± 0.05γ 1.8± 0.1H 0.75–0.80β 0.50–0.70ϕ 1.05± 0.05

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 16/121

Kind of a mess...

Order of business:

1. Find out how these relationships are connected.2. Determine most fundamental description.3. Explain origins of these parameter values

For (3): Many attempts: not yet sorted out...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 18/121

Stream Ordering:

Method for describing network architecture:

I Introduced by Horton (1945) [7]

I Modified by Strahler (1957) [16]

I Term: Horton-Strahler Stream Ordering [11]

I Can be seen as iterative trimming of a network.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 19/121

Stream Ordering:

Some definitions:I A channel head is a point in landscape where flow

becomes focused enough to form a stream.I A source stream is defined as the stream that

reaches from a channel head to a junction withanother stream.

I Roughly analogous to capillary vessels.I Use symbol ω = 1, 2, 3, ... for stream order.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 20/121

Stream Ordering:

1. Label all source streams as order ω = 1 and remove.2. Label all new source streams as order ω = 2 and

remove.3. Repeat until one stream is left (order = Ω)4. Basin is said to be of the order of the last stream

removed.5. Example above is a basin of order Ω = 3.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 21/121

Stream Ordering—A large example:

−105 −100 −95 −90 −8530

32

34

36

38

40

42

44

46

48

longitude

latit

ude

ω = 1110 9 8

Mississippi

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

orde

r_pa

ths_

mis

pi10

.ps]

[21−

Mar

−20

00 p

eter

dod

ds]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 22/121

Stream Ordering:

Another way to define ordering:

I As before, label all source streams as order ω = 1.I Follow all labelled streams downstreamI Whenever two streams of the same order (ω) meet,

the resulting stream has order incremented by 1(ω + 1).

I If streams of different ordersω1 and ω2 meet, then theresultant stream has orderequal to the largest of the two.

I Simple rule:

ω3 = max(ω1, ω2) + δω1,ω2

where δ is the Kronecker delta.−105 −100 −95 −90 −85

30

32

34

36

38

40

42

44

46

48

longitude

latit

ude

ω = 1110 9 8

Mississippi

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

orde

r_pa

ths_

mis

pi10

.ps]

[21−

Mar

−20

00 p

eter

dod

ds]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 23/121

Stream Ordering:

One problem:

I Resolution of data messes with orderingI Micro-description changes (e.g., order of a basin

may increase)I ... but relationships based on ordering appear to be

robust to resolution changes.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 24/121

Stream Ordering:

Utility:

I Stream ordering helpfully discretizes a network.I Goal: understand network architecture

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 25/121

Stream Ordering:

Resultant definitions:I A basin of order Ω has nω streams (or sub-basins) of

order ω.I nω > nω+1

I An order ω basin has area aω.I An order ω basin has a main stream length `ω.I An order ω basin has a stream segment length sω

1. an order ω stream segment is only that part of thestream which is actually of order ω

2. an order ω stream segment runs from the basinoutlet up to the junction of two order ω − 1 streams

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 27/121

Horton’s lawsSelf-similarity of river networks

I First quantified by Horton (1945) [7], expanded bySchumm (1956) [14]

Three laws:I Horton’s law of stream numbers:

nω/nω+1 = Rn > 1

I Horton’s law of stream lengths:

¯ω+1/¯

ω = R` > 1

I Horton’s law of basin areas:

aω+1/aω = Ra > 1

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 28/121

Horton’s laws

Horton’s Ratios:I So... Horton’s laws are defined by three ratios:

Rn, R`, and Ra.

I Horton’s laws describe exponential decay or growth:

nω = nω−1/Rn

= nω−2/R 2n

...

= n1/R ω−1n

= n1e−(ω−1) ln Rn

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 29/121

Horton’s laws

Similar story for area and length:

I

aω = a1e(ω−1) ln Ra

I

¯ω = ¯1e(ω−1) ln R`

I As stream order increases, number drops and areaand length increase.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 30/121

Horton’s laws

A few more things:

I Horton’s laws are laws of averages.I Averaging for number is across basins.I Averaging for stream lengths and areas is within

basins.I Horton’s ratios go a long way to defining a branching

network...I But we need one other piece of information...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 31/121

Horton’s laws

A bonus law:I Horton’s law of stream segment lengths:

sω+1/sω = Rs > 1

I Can show that Rs = R`.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 32/121

Horton’s laws in the real world:

1 2 3 4 5 6 7 8 9 10 11

0

1

2

3

4

5

6

7

ω

(a)

The Mississippi

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

nalo

meg

a_m

ispi

10.p

s]

[15−

Sep

−20

00 p

eter

dod

ds]

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/

dodd

s/w

ork/

river

s/de

ms/

HY

DR

O1K

/afr

ica/

nile

/figu

res/

figna

lom

ega_

nile

.ps]

[10−

Dec

−19

99 p

eter

dod

ds]

1 2 3 4 5 6 7 8 9 10 1110

0

101

102

103

104

105

106

107

stream order ω

The Amazon

nω

aω (sq km)

lω (km)

[sou

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/am

azon

/figu

res/

figna

lom

ega_

amaz

on.p

s]

[16−

Nov

−19

99 p

eter

dod

ds]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 33/121

Horton’s laws-at-large

Blood networks:I Horton’s laws hold for sections of cardiovascular

networksI Measuring such networks is tricky and messy...I Vessel diameters obey an analogous Horton’s law.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 34/121

Horton’s laws

Observations:I Horton’s ratios vary:

Rn 3.0–5.0Ra 3.0–6.0R` 1.5–3.0

I No accepted explanation for these values.I Horton’s laws tell us how quantities vary from level to

level ...I ... but they don’t explain how networks are

structured.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 36/121

Tokunaga’s law

Delving deeper into network architecture:

I Tokunaga (1968) identified a clearer picture ofnetwork structure [21, 22, 23]

I As per Horton-Strahler, use stream ordering.I Focus: describe how streams of different orders

connect to each other.I Tokunaga’s law is also a law of averages.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 37/121

Network Architecture

Definition:I Tµ,ν = the average number of side streams of order

ν that enter as tributaries to streams of order µ

I µ, ν = 1, 2, 3, . . .I µ ≥ ν + 1I Recall each stream segment of order µ is ‘generated’

by two streams of order µ− 1I These generating streams are not considered side

streams.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 38/121

Network ArchitectureTokunaga’s law

I Property 1: Scale independence—depends only ondifference between orders:

Tµ,ν = Tµ−ν

I Property 2: Number of side streams growsexponentially with difference in orders:

Tµ,ν = T1(RT )µ−ν−1

I We usually write Tokunaga’s law as:

Tk = T1(RT )k−1 where RT ' 2

.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 39/121

Tokunaga’s law—an example:

T1 ' 2

RT ' 4

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 40/121

The Mississippi

A Tokunaga graph:

2 3 4 5 6 7 8 9 1011−0.5

0.5

1.5

2.5

µ

log 10

⟨ T

µ,ν ⟩

ν=12 3 4 5 6 7 8 9 10

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 42/121

Can Horton and Tokunaga be happy?

Horton and Tokunaga seem different:

I Horton’s laws appear to contain less detailedinformation than Tokunaga’s law.

I Oddly, Horton’s law has three parameters andTokunaga has two parameters.

I Rn, R`, and Rs versus T1 and RT .I To make a connection, clearest approach is to start

with Tokunaga’s law...I Known result: Tokunaga → Horton [21, 22, 23, 10, 2]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 43/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 44/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 45/121


Putting things together:

I

nω = 2nω+1︸︷︷︸generation

+Ω∑

ω′=ω+1

Tω′−ωnω′︸︷︷︸absorption

I Substitute in Tω′−ω = T1(RT ) ω′−ω−1:

nω = 2nω+1 +Ω∑

ω′=ω+1

T1(RT ) ω′−ω−1nω′

I Shift index to k = ω′ − ω:

nω = 2nω+1 +Ω−ω∑k=1

T1(RT )k−1nω+k

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 46/121


Create Horton ratios:I Divide through by nω+1:

nω

nω+1=

2nω+1

nω+1+

Ω−ω∑k=1

T1(RT )k−1 nω+k

nω+1

I Left hand side looks good but we have nω+k/nω+1’shanging around on the right.

I Recall, we want to show Rn = nω/nω+1 is a constant,independent of ω...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 47/121


Finding Horton ratios:

I Letting Ω→∞, we have

nω

nω+1= 2 +

∞∑k=1

T1(RT )k−1 nω+k

nω+1(1)

I The ratio nω+k/nω+1 can only be a function of k dueto self-similarity (which is implicit in Tokunaga’s law).

I The ratio nω/nω+1 is independent of ω and dependsonly on T1 and RT .

I Can now call nω/nω+1 = Rn.

I Immediately have nω+k/nω+1 = R−(k−1)n .

I Plug into Eq. (1)...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 48/121


Finding Horton ratios:

I Now have:

Rn = 2 +∞∑

k=1

T1(RT )k−1R−(k−1)n

= 2 + T1

∞∑k=1

(RT /Rn)k−1

= 2 + T11

1− RT /Rn

I Rearrange to find:

(Rn − 2)(1− RT /Rn) = T1

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 49/121


Finding Rn in terms of T1 and RT :

I We are here: (Rn − 2)(1− RT /Rn) = T1

I ×Rn to find quadratic in Rn:

(Rn − 2)(Rn − RT ) = T1Rn

I

R 2n − (2 + RT + T1)Rn + 2RT = 0

I Solution:

Rn =(2 + RT + T1)±

√(2 + RT + T1)2 − 8RT

2

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 50/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 51/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 52/121


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 statisticaldistributions. [3, 4]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 53/121


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)...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 54/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 55/121


. . . 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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 56/121


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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 58/121

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

[sou

rce=

/dat

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10.p

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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

[sou

rce=

/dat

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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

aω (sq km)

lω (km)

[sou

rce=

/dat

a11/

dodd

s/w

ork/

river

s/de

ms/

HY

DR

O1K

/afr

ica/

nile

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figna

lom

ega_

nile

.ps]

[10−

Dec

−19

99 p

eter

dod

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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

rce=

/dat

a11/

dodd

s/w

ork/

river

s/de

ms/

HY

DR

O1K

/afr

ica/

nile

/figu

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fignf

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meg

a_ni

le.p

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[10−

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−19

99 p

eter

dod

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

I Highly suggestive that Rn ≡ Ra...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 59/121

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 60/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 61/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 62/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 63/121


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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 64/121


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 Problem set 1 question....

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 65/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 66/121

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

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

equi

part

_mis

pi.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

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

odds

/wor

k/riv

ers/

dem

s/am

azon

/figu

res/

figeq

uipa

rt_a

maz

on.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

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

ica/

nile

/figu

res/

figeq

uipa

rt_n

ile.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 68/121

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)

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 69/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 70/121

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) [24]

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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 71/121

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 [20, 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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 72/121

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 73/121

Scaling laws

Finding 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 `∗

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 74/121

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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 75/121

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 )

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 76/121

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)

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 77/121

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

i=0 an = (ai+1 − 1)/(a− 1)

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 78/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 79/121

Scaling laws

Finding 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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 80/121

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

I Such connections between exponents are calledscaling relations

I Let’s connect to one last relationship: Hack’s law

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 81/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 82/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 83/121

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

rce=

/dat

a6/d

odds

/wor

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

equi

part

_mis

pi.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

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

odds

/wor

k/riv

ers/

dem

s/am

azon

/figu

res/

figeq

uipa

rt_a

maz

on.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

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

ica/

nile

/figu

res/

figeq

uipa

rt_n

ile.p

s]

[15−

Dec

−20

00 p

eter

dod

ds]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 84/121

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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 86/121

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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 87/121

A toy model—Scheidegger’s model

Directed random networks [12, 13]

I

I

P() = P() = 1/2

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 88/121

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−

Dec

−19

99 p

eter

dod

ds]

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 89/121


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

ssip

pi/fi

gure

s/fig

lw_b

low

nup.

ps]

[10−

Dec

−19

99 p

eter

dod

ds]

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 90/121


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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 91/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 92/121


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

k/riv

ers/

dem

s/m

issi

ssip

pi/fi

gure

s/fig

ellw

_col

laps

e_m

ispi

2a.p

s]

[07−

Dec

−20

00 p

eter

dod

ds]

N(s|ω) =1

Rωn Rω

`

F (s/Rω` )

F (x) = e−x/ξ

Mississippi: ξ ' 900 m.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 93/121


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

ssip

pi/fi

gure

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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 94/121


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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 95/121

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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 96/121


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...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 97/121


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).

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 98/121


Network architecture:

I Inter-tributarylengthsexponentiallydistributed

I Leads to randomspatial distributionof streamsegments

1 2

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 99/121


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

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 100/121


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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 101/121


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

.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 102/121


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))

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 103/121



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 )

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 104/121



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) )

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 105/121



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) )

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 107/121

Models

Random subnetworks on a Bethe lattice [15]

I Dominant theoretical conceptfor several decades.

I Bethe lattices are fun andtractable.

I Led to idea of “Statisticalinevitability” of river networkstatistics [8]

I But Bethe latticesunconnected with surfaces.

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

I So let’s move on...

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 108/121

Scheidegger’s model

Directed random networks [12, 13]

I

I

P() = P() = 1/2

I Functional form of all scaling laws exhibited butexponents differ from real world [18, 19, 17]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 109/121

A toy model—Scheidegger’s model

Random walk basins:I Boundaries of basins are random walks

n

x

area a

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 110/121


n

2

6 6

8 8 8 8

9 9Increasing partition of N=64

x

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 111/121


Prob for first return of a random walk in (1+1)dimensions:

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 112/121

Optimal channel networks

Rodríguez-Iturbe, Rinaldo, et al. [11]

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 [9]

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 113/121

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).

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 114/121

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 ()

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 115/121

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.

R. E. Horton.Erosional development of streams and their drainagebasins; hydrophysical approach to quatitativemorphology.Bulletin of the Geological Society of America,56(3):275–370, 1945.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 116/121

References III

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

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 117/121

References IV

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

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..

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 118/121

References V

S. A. Schumm.Evolution of drainage systems and slopes inbadlands at Perth Amboy, New Jersey.Bulletin of the Geological Society of America,67:597–646, May 1956.

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

A. N. Strahler.Hypsometric (area altitude) analysis of erosionaltopography.Bulletin of the Geological Society of America,63:1117–1142, 1952..

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 119/121

References VI

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

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.

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 120/121

References VII

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.

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, 13:1–27, 1978..

BranchingNetworks

Introduction


Allometry

Laws

Stream Ordering

Horton’s Laws

Tokunaga’s Law

Horton⇔ Tokunaga

Reducing Horton

Scaling relations

Fluctuations

Models

References

Frame 121/121

References VIII

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

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

outline branching networks introduction

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