Download - 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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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/ ()
BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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.
BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
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longitude
latit
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ω = 1110 9 8
Mississippi
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BranchingNetworks
Introduction
River NetworksDefinitions
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
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longitude
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ω = 1110 9 8
Mississippi
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.ps]
[21−
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dod
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BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
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ω
(a)
The Mississippi
[sou
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issi
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s/fig
nalo
meg
a_m
ispi
10.p
s]
[15−
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1 2 3 4 5 6 7 8 9 10 1110
−1
100
101
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103
104
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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−
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−19
99 p
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1 2 3 4 5 6 7 8 9 10 1110
0
101
102
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106
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stream order ω
The Amazon
nω
aω (sq km)
lω (km)
[sou
rce=
/dat
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odds
/wor
k/riv
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dem
s/am
azon
/figu
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figna
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amaz
on.p
s]
[16−
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99 p
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BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 45/121
More with the happy-making thing
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 46/121
More with the happy-making thing
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 47/121
More with the happy-making thing
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 48/121
More with the happy-making thing
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 49/121
More with the happy-making thing
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 52/121
Horton and Tokunaga are happy
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 53/121
Horton and Tokunaga are happy
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 55/121
Horton and Tokunaga are friends
. . . 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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 56/121
Horton and Tokunaga are friends
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
River NetworksDefinitions
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
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−6
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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)
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1 2 3 4 5 6 7 8 9 10 1110
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100
101
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103
104
105
106
107
stream order ω
The Nile
nΩ−ω+1aω (sq km)
Ω = 10
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I In right plots, stream number graph has been flippedvertically.
I Highly suggestive that Rn ≡ Ra...
BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 63/121
Reducing Horton’s laws:
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 64/121
Reducing Horton’s laws:
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
River NetworksDefinitions
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
River NetworksDefinitions
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
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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
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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
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BranchingNetworks
Introduction
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
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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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
)
Mississippi: length distributions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 89/121
Generalizing Horton’s laws
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/
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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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 90/121
Generalizing Horton’s laws
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 92/121
Generalizing Horton’s laws
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
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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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 93/121
Generalizing Horton’s laws
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)
Mississippi: length distributions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 94/121
Generalizing Horton’s laws
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 96/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 97/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 98/121
Generalizing Tokunaga’s law
Network architecture:
I Inter-tributarylengthsexponentiallydistributed
I Leads to randomspatial distributionof streamsegments
1 2
BranchingNetworks
Introduction
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 99/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 100/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 101/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 102/121
Generalizing Tokunaga’s law
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 103/121
Generalizing Tokunaga’s law
I Checking form of P(sµ, Tµ,ν) works:
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 104/121
Generalizing Tokunaga’s law
I Checking form of P(sµ, Tµ,ν) works:
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 105/121
Generalizing Tokunaga’s law
I Checking form of P(sµ, Tµ,ν) works:
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 110/121
Scheidegger’s model
n
2
6 6
8 8 8 8
9 9Increasing partition of N=64
x
BranchingNetworks
Introduction
River NetworksDefinitions
Allometry
Laws
Stream Ordering
Horton’s Laws
Tokunaga’s Law
Horton⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
References
Frame 111/121
Scheidegger’s model
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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
River NetworksDefinitions
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.