Branching Networks II Horton Tokunaga€¦ · Branching Networks II Horton ⇔ Tokunaga Reducing...
Transcript of Branching Networks II Horton Tokunaga€¦ · Branching Networks II Horton ⇔ Tokunaga Reducing...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 1/74
Branching Networks IIComplex Networks, Course 303A, Spring, 2009
Prof. Peter Dodds
Department of Mathematics & StatisticsUniversity of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 2/74
Outline
Horton ⇔ Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 3/74
Can Horton and Tokunaga be happy?
Horton and Tokunaga seem different:
I In terms of network achitecture, Horton’s lawsappear to contain less detailed information thanTokunaga’s law.
I Oddly, Horton’s laws have four parameters andTokunaga has two parameters.
I Rn, Ra, R`, and Rs versus T1 and RT . One simpleredundancy: R` = Rs.Insert question from assignment 1 ()
I To make a connection, clearest approach is to startwith Tokunaga’s law...
I Known result: Tokunaga → Horton [18, 19, 20, 9, 2]
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 4/74
Let us make them happy
We need one more ingredient:
Space-fillingness
I A network is space-filling if the average distancebetween adjacent streams is roughly constant.
I Reasonable for river and cardiovascular networksI For river networks:
Drainage density ρdd = inverse of typical distancebetween channels in a landscape.
I In terms of basin characteristics:
ρdd '∑
stream segment lengthsbasin area
=
∑Ωω=1 nωsω
aΩ
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 5/74
More with the happy-making thing
Start with Tokunaga’s law: Tk = T1Rk−1T
I Start looking for Horton’s stream number law:nω/nω+1 = Rn.
I Estimate nω, the number of streams of order ω interms of other nω′ , ω′ > ω.
I Observe that each stream of order ω terminates byeither:
ω=3
ω=4
ω=3
ω=3
ω=4
ω=4
1. Running into another stream of order ωand generating a stream of order ω + 1...
I 2nω+1 streams of order ω do this
2. Running into and being absorbed by astream of higher order ω′ > ω...
I n′ωTω′−ω streams of order ω do this
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 6/74
More with the happy-making thing
Putting things together:
I
nω = 2nω+1︸ ︷︷ ︸generation
+Ω∑
ω′=ω+1
Tω′−ωnω′︸ ︷︷ ︸absorption
I Use Tokunaga’s law and manipulate expression tocreate Rn’s.
I Insert question from assignment 1 ()I Solution:
Rn =(2 + RT + T1)±
√(2 + RT + T1)2 − 8RT
2
(The larger value is the one we want.)
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 7/74
Finding other Horton ratios
Connect Tokunaga to Rs
I Now use uniform drainage density ρdd.I Assume side streams are roughly separated by
distance 1/ρdd.I For an order ω stream segment, expected length is
sω ' ρ−1dd
(1 +
ω−1∑k=1
Tk
)
I Substitute in Tokunaga’s law Tk = T1Rk−1T :
sω ' ρ−1dd
(1 + T1
ω−1∑k=1
R k−1T
)∝ R ω
T
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 8/74
Horton and Tokunaga are happy
Altogether then:
I
⇒ sω/sω−1 = RT ⇒ Rs = RT
I Recall R` = Rs so
R` = RT
I And from before:
Rn =(2 + RT + T1) +
√(2 + RT + T1)2 − 8RT
2
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 9/74
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 non-trivialstatistical distributions. [3, 4]
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 10/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 11/74
Horton and Tokunaga are friends
From Horton to Tokunaga [2]
(Rl)(a)(b)(c)
I Assume Horton’s lawshold for number andlength
I Start with an order ωstream
I Scale up by a factor ofR`, orders increment
I Maintain drainagedensity by adding neworder 1 streams
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 12/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 13/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 14/74
Horton’s laws of area and number:
1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
4
5
6
7
ω
(a)
The Mississippi
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1 2 3 4 5 6 7 8 9 10 11−7
−6
−5
−4
−3
−2
−1
0
ω
(b)
The Mississippi
Ω = 11
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1 2 3 4 5 6 7 8 9 10 1110
−1
100
101
102
103
104
105
106
107
stream order ω
The Nile
nω
aω (sq km)
lω (km)
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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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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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 15/74
Measuring Horton ratios is tricky:
I How robust are our estimates of ratios?I Rule of thumb: discard data for two smallest and two
largest orders.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 16/74
Mississippi:
ω range Rn Ra R` Rs Ra/Rn[2, 3] 5.27 5.26 2.48 2.30 1.00[2, 5] 4.86 4.96 2.42 2.31 1.02[2, 7] 4.77 4.88 2.40 2.31 1.02[3, 4] 4.72 4.91 2.41 2.34 1.04[3, 6] 4.70 4.83 2.40 2.35 1.03[3, 8] 4.60 4.79 2.38 2.34 1.04[4, 6] 4.69 4.81 2.40 2.36 1.02[4, 8] 4.57 4.77 2.38 2.34 1.05[5, 7] 4.68 4.83 2.36 2.29 1.03[6, 7] 4.63 4.76 2.30 2.16 1.03[7, 8] 4.16 4.67 2.41 2.56 1.12
mean µ 4.69 4.85 2.40 2.33 1.04std dev σ 0.21 0.13 0.04 0.07 0.03
σ/µ 0.045 0.027 0.015 0.031 0.024
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 17/74
Amazon:
ω range Rn Ra R` Rs Ra/Rn[2, 3] 4.78 4.71 2.47 2.08 0.99[2, 5] 4.55 4.58 2.32 2.12 1.01[2, 7] 4.42 4.53 2.24 2.10 1.02[3, 5] 4.45 4.52 2.26 2.14 1.01[3, 7] 4.35 4.49 2.20 2.10 1.03[4, 6] 4.38 4.54 2.22 2.18 1.03[5, 6] 4.38 4.62 2.22 2.21 1.06[6, 7] 4.08 4.27 2.05 1.83 1.05
mean µ 4.42 4.53 2.25 2.10 1.02std dev σ 0.17 0.10 0.10 0.09 0.02
σ/µ 0.038 0.023 0.045 0.042 0.019
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 18/74
Reducing Horton’s laws:
Rough first effort to show Rn ≡ Ra:
I aΩ ∝ sum of all stream lengths in a order Ω basin(assuming uniform drainage density)
I So:
aΩ 'Ω∑
ω=1
nωsω/ρdd
∝Ω∑
ω=1
R Ω−ωn ·
nΩ︷︸︸︷1︸ ︷︷ ︸
nω
s1 · R ω−1s︸ ︷︷ ︸
sω
=R Ω
nRs
s1
Ω∑ω=1
(Rs
Rn
)ω
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 19/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 20/74
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 Insert question from assignment 1 ()
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 21/74
Equipartitioning:
Intriguing division of area:
I Observe: Combined area of basins of order ωindependent of ω.
I Not obvious: basins of low orders not necessarilycontained in basis on higher orders.
I Story:Rn ≡ Ra ⇒ nωaω = const
I Reason:nω ∝ (Rn)
−ω
aω ∝ (Ra)ω ∝ n−1
ω
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 22/74
Equipartitioning:Some examples:
1 2 3 4 5 6 7 8 9 10 110
0.2
0.4
0.6
0.8
1
ω
n ω a
ω /
a Ω
Mississippi basin partitioning
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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
[sou
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O1K
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BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 23/74
Scaling laws
The story so far:
I Natural branching networks are hierarchical,self-similar structures
I Hierarchy is mixedI Tokunaga’s law describes detailed architecture:
Tk = T1Rk−1T .
I We have connected Tokunaga’s and Horton’s lawsI Only two Horton laws are independent (Rn = Ra)I Only two parameters are independent:
(T1, RT )⇔ (Rn, Rs)
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 24/74
Scaling laws
A little further...I Ignore stream ordering for the momentI Pick a random location on a branching network p.I Each point p is associated with a basin and a longest
stream lengthI Q: What is probability that the p’s drainage basin has
area a? P(a) ∝ a−τ for large aI Q: What is probability that the longest stream from p
has length `? P(`) ∝ `−γ for large `
I Roughly observed: 1.3 . τ . 1.5 and 1.7 . γ . 2.0
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 25/74
Scaling laws
Probability distributions with power-law decays
I We see them everywhere:I Earthquake magnitudes (Gutenberg-Richter law)I City sizes (Zipf’s law)I Word frequency (Zipf’s law) [21]
I Wealth (maybe not—at least heavy tailed)I Statistical mechanics (phase transitions) [5]
I A big part of the story of complex systemsI Arise from mechanisms: growth, randomness,
optimization, ...I Our task is always to illuminate the mechanism...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 26/74
Scaling laws
Connecting exponents
I We have the detailed picture of branching networks(Tokunaga and Horton)
I Plan: Derive P(a) ∝ a−τ and P(`) ∝ `−γ starting withTokunaga/Horton story [17, 1, 2]
I Let’s work on P(`)...I Our first fudge: assume Horton’s laws hold
throughout a basin of order Ω.I (We know they deviate from strict laws for low ω and
high ω but not too much.)I Next: place stick between teeth. Bite stick. Proceed.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 27/74
Scaling laws
Finding γ:
I Often useful to work with cumulative distributions,especially when dealing with power-law distributions.
I The complementary cumulative distribution turns outto be most useful:
P>(`∗) = P(` > `∗) =
∫ `max
`=`∗
P(`)d`
I
P>(`∗) = 1− P(` < `∗)
I Also known as the exceedance probability.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 28/74
Scaling lawsFinding γ:
I The connection between P(x) and P>(x) when P(x)has a power law tail is simple:
I Given P(`) ∼ `−γ large ` then for large enough `∗
P>(`∗) =
∫ `max
`=`∗
P(`) d`
∼∫ `max
`=`∗
`−γd`
=`−γ+1
−γ + 1
∣∣∣∣`max
`=`∗
∝ `−γ+1∗ for `max `∗
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 29/74
Scaling laws
Finding γ:
I Aim: determine probability of randomly choosing apoint on a network with main stream length > `∗
I Assume some spatial sampling resolution ∆
I Landscape is broken up into grid of ∆×∆ sitesI Approximate P>(`∗) as
P>(`∗) =N>(`∗;∆)
N>(0;∆).
where N>(`∗;∆) is the number of sites with mainstream length > `∗.
I Use Horton’s law of stream segments:sω/sω−1 = Rs...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 30/74
Scaling laws
Finding γ:
I Set `∗ = `ω for some 1 ω Ω.I
P>(`ω) =N>(`ω;∆)
N>(0;∆)'∑Ω
ω′=ω+1 nω′sω′/∆∑Ωω′=1 nω′sω′/∆
I ∆’s cancelI Denominator is aΩρdd, a constant.I So... using Horton’s laws...
P>(`ω) ∝Ω∑
ω′=ω+1
nω′sω′ 'Ω∑
ω′=ω+1
(1·R Ω−ω′n )(s1·R ω′−1
s )
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 31/74
Scaling laws
Finding γ:
I We are here:
P>(`ω) ∝Ω∑
ω′=ω+1
(1 · R Ω−ω′n )(s1 · R ω′−1
s )
I Cleaning up irrelevant constants:
P>(`ω) ∝Ω∑
ω′=ω+1
(Rs
Rn
)ω′
I Change summation order by substitutingω′′ = Ω− ω′.
I Sum is now from ω′′ = 0 to ω′′ = Ω− ω − 1(equivalent to ω′ = Ω down to ω′ = ω + 1)
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 32/74
Scaling laws
Finding γ:
I
P>(`ω) ∝Ω−ω−1∑ω′′=0
(Rs
Rn
)Ω−ω′′
∝Ω−ω−1∑ω′′=0
(Rn
Rs
)ω′′
I Since Rn > Rs and 1 ω Ω,
P>(`ω) ∝(
Rn
Rs
)Ω−ω
∝(
Rn
Rs
)−ω
again using∑n−1
i=0 ai = (a n − 1)/(a− 1)
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 33/74
Scaling laws
Finding γ:
I Nearly there:
P>(`ω) ∝(
Rn
Rs
)−ω
= e−ω ln(Rn/Rs)
I Need to express right hand side in terms of `ω.I Recall that `ω ' ¯1R ω−1
` .I
`ω ∝ R ω` = R ω
s = e ω ln Rs
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 34/74
Scaling lawsFinding γ:
I Therefore:
P>(`ω) ∝ e−ω ln(Rn/Rs) =(
e ω ln Rs)− ln(Rn/Rs)/ ln(Rs)
I
∝ `ω− ln(Rn/Rs)/ ln Rs
I
= `−(ln Rn−ln Rs)/ ln Rsω
I
= `− ln Rn/ ln Rs+1ω
I
= `−γ+1ω
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 35/74
Scaling laws
Finding γ:
I And so we have:
γ = ln Rn/ ln Rs
I Proceeding in a similar fashion, we can show
τ = 2− ln Rs/ ln Rn = 2− 1/γ
Insert question from assignment 1 ()I Such connections between exponents are called
scaling relationsI Let’s connect to one last relationship: Hack’s law
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 36/74
Scaling laws
Hack’s law: [6]
I
` ∝ ah
I Typically observed that 0.5 . h . 0.7.I Use Horton laws to connect h to Horton ratios:
`ω ∝ R ωs and aω ∝ R ω
n
I Observe:
`ω ∝ e ω ln Rs ∝(
e ω ln Rn)ln Rs/ ln Rn
∝ (R ωn )ln Rs/ ln Rn ∝ a ln Rs/ ln Rn
ω ⇒ h = ln Rs/ ln Rn
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 37/74
Connecting exponentsOnly 3 parameters are independent:e.g., take d , Rn, and Rs
relation: scaling relation/parameter: [2]
` ∼ Ld dTk = T1(RT )k−1 T1 = Rn − Rs − 2 + 2Rs/Rn
RT = Rsnω/nω+1 = Rn Rnaω+1/aω = Ra Ra = Rn¯ω+1/¯
ω = R` R` = Rs` ∼ ah h = log Rs/ log Rna ∼ LD D = d/h
L⊥ ∼ LH H = d/h − 1P(a) ∼ a−τ τ = 2− hP(`) ∼ `−γ γ = 1/h
Λ ∼ aβ β = 1 + hλ ∼ Lϕ ϕ = d
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 38/74
Equipartitioning reexamined:Recall this story:
1 2 3 4 5 6 7 8 9 10 110
0.2
0.4
0.6
0.8
1
ω
n ω a
ω /
a Ω
Mississippi basin partitioning
[sou
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 39/74
Equipartitioning
I What aboutP(a) ∼ a−τ ?
I Since τ > 1, suggests no equipartitioning:
aP(a) ∼ a−τ+1 6= const
I P(a) overcounts basins within basins...I while stream ordering separates basins...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 40/74
Fluctuations
Moving beyond the mean:
I Both Horton’s laws and Tokunaga’s law relateaverage properties, e.g.,
sω/sω−1 = Rs
I Natural generalization to consideration relationshipsbetween probability distributions
I Yields rich and full description of branching networkstructure
I See into the heart of randomness...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 41/74
A toy model—Scheidegger’s model
Directed random networks [11, 12]
I
I
P() = P() = 1/2
I Flow is directed downwardsI Useful and interesting test case—more later...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 42/74
Generalizing Horton’s laws
I ¯ω ∝ (R`)
ω ⇒ N(`|ω) = (RnR`)−ωF`(`/Rω
` )
I aω ∝ (Ra)ω ⇒ N(a|ω) = (R2
n)−ωFa(a/Rωn )
0 100 200 300 40010
−4
10−2
100
102
l (km)
N(l
|ω)
Mississippi: length distributions
ω=3 4 5 6
[sou
rce=
/dat
a6/d
odds
/wor
k/riv
ers/
dem
s/m
issi
ssip
pi/fi
gure
s/fig
lw_c
olla
pse_
mis
pi2.
ps]
[09−
Dec
−19
99 p
eter
dod
ds]
0 1 2 310
−7
10−6
10−5
10−4
10−3
l Rl−ω
Rnω
−Ω
Rlω N
(l |ω
)
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 43/74
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/
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 44/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 45/74
Combining stream segments distributions:
I Stream segmentssum to give mainstream lengths
I
`ω =
µ=ω∑µ=1
sµ
I P(`ω) is aconvolution ofdistributions forthe sω
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 46/74
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
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 47/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 48/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 49/74
Generalizing Tokunaga’s law
Scheidegger:
0 100 200 300−5
−4
−3
−2
−1
(a)
Tµ,ν
log 10
P(T
µ,ν )
0 0.1 0.2 0.3 0.4 0.5 0.6−3
−2
−1
0
1(b)
Tµ,ν (Rl (s))−µ
log 10
(Rl (
s) )
µ P(T
µ,ν )
I Observe exponential distributions for Tµ,ν
I Scaling collapse works using Rs
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 50/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 51/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 52/74
Generalizing Tokunaga’s law
Network architecture:
I Inter-tributarylengthsexponentiallydistributed
I Leads to randomspatial distributionof streamsegments
1 2
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 53/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 54/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 55/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 56/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 57/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 58/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 59/74
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 II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 60/74
Models
Random subnetworks on a Bethe lattice [13]
I Dominant theoretical conceptfor several decades.
I Bethe lattices are fun andtractable.
I Led to idea of “Statisticalinevitability” of river networkstatistics [7]
I But Bethe latticesunconnected with surfaces.
I In fact, Bethe lattices 'infinite dimensional spaces(oops).
I So let’s move on...
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 61/74
Scheidegger’s model
Directed random networks [11, 12]
I
I
P() = P() = 1/2
I Functional form of all scaling laws exhibited butexponents differ from real world [15, 16, 14]
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 62/74
A toy model—Scheidegger’s model
Random walk basins:I Boundaries of basins are random walks
n
x
area a
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 63/74
Scheidegger’s model
n
2
6 6
8 8 8 8
9 9Increasing partition of N=64
x
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 64/74
Scheidegger’s model
Prob for first return of a random walk in (1+1)dimensions (from CSYS/MATH 300):
I
P(n) ∼ 12√
πn−3/2.
and so P(`) ∝ `−3/2.I Typical area for a walk of length n is ∝ n3/2:
` ∝ a 2/3.
I Find τ = 4/3, h = 2/3, γ = 3/2, d = 1.I Note τ = 2− h and γ = 1/h.I Rn and R` have not been derived analytically.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 65/74
Optimal channel networks
Rodríguez-Iturbe, Rinaldo, et al. [10]
I Landscapes h(~x) evolve such that energy dissipationε is minimized, where
ε ∝∫
d~r (flux)× (force) ∼∑
i
ai∇hi ∼∑
i
aγi
I Landscapes obtained numerically give exponentsnear that of real networks.
I But: numerical method used matters.I And: Maritan et al. find basic universality classes are
that of Scheidegger, self-similar, and a third kind ofrandom network [8]
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 66/74
Theoretical networks
Summary of universality classes:
network h dNon-convergent flow 1 1
Directed random 2/3 1Undirected random 5/8 5/4
Self-similar 1/2 1OCN’s (I) 1/2 1OCN’s (II) 2/3 1OCN’s (III) 3/5 1Real rivers 0.5–0.7 1.0–1.2
h ⇒ ` ∝ ah (Hack’s law).d ⇒ ` ∝ Ld
‖ (stream self-affinity).
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 67/74
Nutshell
Branching networks II Key Points:
I Horton’s laws and Tokunaga law all fit together.I nb. for 2-d networks, these laws are ‘planform’ laws
and ignore slope.I Abundant scaling relations can be derived.I Can take Rn, R`, and d as three independent
parameters necessary to describe all 2-d branchingnetworks.
I For scaling laws, only h = ln R`/ ln Rn and d areneeded.
I Laws can be extended nicely to laws of distributions.I Numerous models of branching network evolution
exist: nothing rock solid yet.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 68/74
References I
H. de Vries, T. Becker, and B. Eckhardt.Power law distribution of discharge in ideal networks.Water Resources Research, 30(12):3541–3543,December 1994.
P. S. Dodds and D. H. Rothman.Unified view of scaling laws for river networks.Physical Review E, 59(5):4865–4877, 1999. pdf ()
P. S. Dodds and D. H. Rothman.Geometry of river networks. II. Distributions ofcomponent size and number.Physical Review E, 63(1):016116, 2001. pdf ()
P. S. Dodds and D. H. Rothman.Geometry of river networks. III. Characterization ofcomponent connectivity.Physical Review E, 63(1):016117, 2001. pdf ()
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 69/74
References II
N. Goldenfeld.Lectures on Phase Transitions and theRenormalization Group, volume 85 of Frontiers inPhysics.Addison-Wesley, Reading, Massachusetts, 1992.
J. T. Hack.Studies of longitudinal stream profiles in Virginia andMaryland.United States Geological Survey Professional Paper,294-B:45–97, 1957.
J. W. Kirchner.Statistical inevitability of Horton’s laws and theapparent randomness of stream channel networks.Geology, 21:591–594, July 1993.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 70/74
References III
A. Maritan, F. Colaiori, A. Flammini, M. Cieplak, andJ. R. Banavar.Universality classes of optimal channel networks.Science, 272:984–986, 1996. pdf ()
S. D. Peckham.New results for self-similar trees with applications toriver networks.Water Resources Research, 31(4):1023–1029, April1995.
I. Rodríguez-Iturbe and A. Rinaldo.Fractal River Basins: Chance and Self-Organization.Cambridge University Press, Cambrigde, UK, 1997.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 71/74
References IV
A. E. Scheidegger.A stochastic model for drainage patterns into anintramontane trench.Bull. Int. Assoc. Sci. Hydrol., 12(1):15–20, 1967.
A. E. Scheidegger.Theoretical Geomorphology.Springer-Verlag, New York, third edition, 1991.
R. L. Shreve.Infinite topologically random channel networks.Journal of Geology, 75:178–186, 1967.
H. Takayasu.Steady-state distribution of generalized aggregationsystem with injection.Physcial Review Letters, 63(23):2563–2565,December 1989.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 72/74
References V
H. Takayasu, I. Nishikawa, and H. Tasaki.Power-law mass distribution of aggregation systemswith injection.Physical Review A, 37(8):3110–3117, April 1988.
M. Takayasu and H. Takayasu.Apparent independency of an aggregation systemwith injection.Physical Review A, 39(8):4345–4347, April 1989.
D. G. Tarboton, R. L. Bras, and I. Rodríguez-Iturbe.Comment on “On the fractal dimension of streamnetworks” by Paolo La Barbera and Renzo Rosso.Water Resources Research, 26(9):2243–4,September 1990.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 73/74
References VI
E. Tokunaga.The composition of drainage network in ToyohiraRiver Basin and the valuation of Horton’s first law.Geophysical Bulletin of Hokkaido University, 15:1–19,1966.
E. Tokunaga.Consideration on the composition of drainagenetworks and their evolution.Geographical Reports of Tokyo MetropolitanUniversity, 13:G1–27, 1978.
E. Tokunaga.Ordering of divide segments and law of dividesegment numbers.Transactions of the Japanese GeomorphologicalUnion, 5(2):71–77, 1984.
BranchingNetworks II
Horton ⇔Tokunaga
Reducing Horton
Scaling relations
Fluctuations
Models
Nutshell
References
Frame 74/74
References VII
G. K. Zipf.Human Behaviour and the Principle of Least-Effort.Addison-Wesley, Cambridge, MA, 1949.