Branching Networks Introduction River Networks Complex Networks, - PowerPoint PPT Presentation

Branching Stream Ordering: Networks Introduction River Networks Definitions Allometry Laws Stream Ordering One problem: Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga ◮ Resolution of data messes with ordering Reducing Horton Scaling relations Fluctuations ◮ Micro-description changes (e.g., order of a basin Models may increase) References ◮ ... but relationships based on ordering appear to be robust to resolution changes. Frame 23/121

Branching Stream Ordering: Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Utility: Reducing Horton Scaling relations Fluctuations ◮ Stream ordering helpfully discretizes a network. Models References ◮ Goal: understand network architecture Frame 24/121

Branching Stream Ordering: Networks Introduction River Networks Definitions Resultant definitions: Allometry Laws ◮ A basin of order Ω has n ω streams (or sub-basins) of Stream Ordering Horton’s Laws order ω . Tokunaga’s Law Horton ⇔ Tokunaga ◮ n ω > n ω + 1 Reducing Horton Scaling relations Fluctuations ◮ An order ω basin has area a ω . Models ◮ An order ω basin has a main stream length ℓ ω . References ◮ An order ω basin has a stream segment length s ω 1. an order ω stream segment is only that part of the stream which is actually of order ω 2. an order ω stream segment runs from the basin outlet up to the junction of two order ω − 1 streams Frame 25/121

Branching Horton’s laws Networks Self-similarity of river networks Introduction ◮ First quantified by Horton (1945) [7] , expanded by River Networks Definitions Schumm (1956) [14] Allometry Laws Stream Ordering Horton’s Laws Three laws: Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Horton’s law of stream numbers: Scaling relations Fluctuations Models n ω / n ω + 1 = R n > 1 References ◮ Horton’s law of stream lengths: ℓ ω + 1 / ¯ ¯ ℓ ω = R ℓ > 1 ◮ Horton’s law of basin areas: a ω + 1 / ¯ ¯ a ω = R a > 1 Frame 27/121

Branching Horton’s laws Networks Horton’s Ratios: Introduction River Networks ◮ So... Horton’s laws are defined by three ratios: Definitions Allometry Laws R n , R ℓ , and R a . Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Horton’s laws describe exponential decay or growth: Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations n ω = n ω − 1 / R n Models References = n ω − 2 / R 2 n . . . = n 1 / R ω − 1 n = n 1 e − ( ω − 1 ) ln R n Frame 28/121

Branching Horton’s laws Networks Introduction River Networks Definitions Allometry Laws Similar story for area and length: Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Horton ⇔ Tokunaga Reducing Horton a 1 e ( ω − 1 ) ln R a a ω = ¯ ¯ Scaling relations Fluctuations Models ◮ References ℓ ω = ¯ ¯ ℓ 1 e ( ω − 1 ) ln R ℓ ◮ As stream order increases, number drops and area and length increase. Frame 29/121

Branching Horton’s laws Networks Introduction River Networks Definitions Allometry A few more things: Laws Stream Ordering Horton’s Laws ◮ Horton’s laws are laws of averages. Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Averaging for number is across basins. Scaling relations Fluctuations ◮ Averaging for stream lengths and areas is within Models References basins. ◮ Horton’s ratios go a long way to defining a branching network... ◮ But we need one other piece of information... Frame 30/121

Branching Horton’s laws Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws A bonus law: Tokunaga’s Law Horton ⇔ Tokunaga ◮ Horton’s law of stream segment lengths: Reducing Horton Scaling relations Fluctuations Models ¯ s ω + 1 / ¯ s ω = R s > 1 References ◮ Can show that R s = R ℓ . Frame 31/121

Branching Horton’s laws in the real world: Networks The Nile The Mississippi 7 7 10 n ω Introduction 6 10 a ω (sq km) 6 River Networks l ω (km) 5 10 5 Definitions (a) [source=/data11/dodds/work/rivers/dems/HYDRO1K/africa/nile/figures/fignalomega_nile.ps] Allometry [source=/data6/dodds/work/rivers/dems/mississippi/figures/fignalomega_mispi10.ps] 4 10 4 Laws 3 Stream Ordering 10 3 Horton’s Laws 2 10 [15−Sep−2000 peter dodds] [10−Dec−1999 peter dodds] Tokunaga’s Law 2 1 Horton ⇔ Tokunaga 10 1 Reducing Horton 0 10 Scaling relations 0 Fluctuations −1 10 Models 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 ω stream order ω References The Amazon 7 10 n ω a ω (sq km) 6 10 l ω (km) 5 10 [source=/data6/dodds/work/rivers/dems/amazon/figures/fignalomega_amazon.ps] 4 10 3 10 [16−Nov−1999 peter dodds] 2 10 1 10 0 10 Frame 32/121 1 2 3 4 5 6 7 8 9 10 11 stream order ω

Branching Horton’s laws-at-large Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Blood networks: Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Horton’s laws hold for sections of cardiovascular Scaling relations Fluctuations networks Models References ◮ Measuring such networks is tricky and messy... ◮ Vessel diameters obey an analogous Horton’s law. Frame 33/121

Branching Horton’s laws Networks Introduction River Networks Definitions Observations: Allometry Laws Stream Ordering ◮ Horton’s ratios vary: Horton’s Laws Tokunaga’s Law R n 3.0–5.0 Horton ⇔ Tokunaga Reducing Horton R a 3.0–6.0 Scaling relations Fluctuations R ℓ 1.5–3.0 Models References ◮ No accepted explanation for these values. ◮ Horton’s laws tell us how quantities vary from level to level ... ◮ ... but they don’t explain how networks are structured. Frame 34/121

Branching Tokunaga’s law Networks Introduction River Networks Definitions Allometry Laws Delving deeper into network architecture: Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Tokunaga (1968) identified a clearer picture of Horton ⇔ Tokunaga Reducing Horton network structure [21, 22, 23] Scaling relations Fluctuations Models ◮ As per Horton-Strahler, use stream ordering. References ◮ Focus: describe how streams of different orders connect to each other. ◮ Tokunaga’s law is also a law of averages. Frame 36/121

Branching Network Architecture Networks Introduction River Networks Definitions Allometry Definition: Laws Stream Ordering ◮ T µ,ν = the average number of side streams of order Horton’s Laws Tokunaga’s Law ν that enter as tributaries to streams of order µ Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ µ , ν = 1, 2, 3, . . . Fluctuations Models ◮ µ ≥ ν + 1 References ◮ Recall each stream segment of order µ is ‘generated’ by two streams of order µ − 1 ◮ These generating streams are not considered side streams. Frame 37/121

Branching Network Architecture Networks Tokunaga’s law Introduction ◮ Property 1: Scale independence—depends only on River Networks Definitions difference between orders: Allometry Laws Stream Ordering Horton’s Laws T µ,ν = T µ − ν Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations Models ◮ Property 2: Number of side streams grows References exponentially with difference in orders: T µ,ν = T 1 ( R T ) µ − ν − 1 ◮ We usually write Tokunaga’s law as: T k = T 1 ( R T ) k − 1 where R T ≃ 2 Frame 38/121 .

Branching Tokunaga’s law—an example: Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations T 1 ≃ 2 Models References R T ≃ 4 Frame 39/121

Branching The Mississippi Networks Introduction A Tokunaga graph: River Networks Definitions Allometry ν =1 Laws Stream Ordering 2 Horton’s Laws 2.5 Tokunaga’s Law 3 Horton ⇔ Tokunaga log 10 〈 T µ , ν 〉 Reducing Horton 4 Scaling relations Fluctuations 5 1.5 Models 6 References 7 8 0.5 9 10 −0.5 2 3 4 5 6 7 8 9 1011 µ Frame 40/121

Branching Can Horton and Tokunaga be happy? Networks Introduction River Networks Definitions Allometry Horton and Tokunaga seem different: Laws Stream Ordering Horton’s Laws ◮ Horton’s laws appear to contain less detailed Tokunaga’s Law Horton ⇔ Tokunaga information than Tokunaga’s law. Reducing Horton Scaling relations ◮ Oddly, Horton’s law has three parameters and Fluctuations Models Tokunaga has two parameters. References ◮ R n , R ℓ , and R s versus T 1 and R T . ◮ To make a connection, clearest approach is to start with Tokunaga’s law... ◮ Known result: Tokunaga → Horton [21, 22, 23, 10, 2] Frame 42/121

Branching Let us make them happy Networks Introduction We need one more ingredient: River Networks Definitions Space-fillingness Allometry Laws Stream Ordering ◮ A network is space-filling if the average distance Horton’s Laws Tokunaga’s Law between adjacent streams is roughly constant. Horton ⇔ Tokunaga Reducing Horton ◮ Reasonable for river and cardiovascular networks Scaling relations Fluctuations Models ◮ For river networks: References Drainage density ρ dd = inverse of typical distance between channels in a landscape. ◮ In terms of basin characteristics: � stream segment lengths � Ω ω = 1 n ω s ω ρ dd ≃ = basin area a Ω Frame 43/121

Branching More with the happy-making thing Networks Start with Tokunaga’s law: T k = T 1 R k − 1 Introduction T River Networks ◮ Start looking for Horton’s stream number law: Definitions Allometry n ω / n ω + 1 = R n . Laws Stream Ordering ◮ Estimate n ω , the number of streams of order ω in Horton’s Laws Tokunaga’s Law terms of other n ω ′ , ω ′ > ω . Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ Observe that each stream of order ω terminates by Fluctuations Models either: References ω=3 ω=3 1. Running into another stream of order ω and generating a stream of order ω + 1... ω=4 ◮ 2 n ω + 1 streams of order ω do this ω=3 2. Running into and being absorbed by a ω=4 stream of higher order ω ′ > ω ... ◮ n ′ ω=4 ω T ω ′ − ω streams of order ω do this Frame 44/121

Branching More with the happy-making thing Networks Putting things together: Introduction River Networks ◮ Definitions Ω Allometry � Laws n ω = 2 n ω + 1 + T ω ′ − ω n ω ′ Stream Ordering � �� � � �� � Horton’s Laws ω ′ = ω + 1 Tokunaga’s Law absorption generation Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ Substitute in T ω ′ − ω = T 1 ( R T ) ω ′ − ω − 1 : Fluctuations Models References Ω � T 1 ( R T ) ω ′ − ω − 1 n ω ′ n ω = 2 n ω + 1 + ω ′ = ω + 1 ◮ Shift index to k = ω ′ − ω : Ω − ω � T 1 ( R T ) k − 1 n ω + k n ω = 2 n ω + 1 + Frame 45/121 k = 1

Branching More with the happy-making thing Networks Introduction River Networks Definitions Allometry Create Horton ratios: Laws Stream Ordering Horton’s Laws ◮ Divide through by n ω + 1 : Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Ω − ω Scaling relations n ω = 2 ✘✘ n ω + 1 T 1 ( R T ) k − 1 n ω + k � ✘ Fluctuations + Models n ω + 1 n ω + 1 n ω + 1 ✘ ✘✘ References k = 1 ◮ Left hand side looks good but we have n ω + k / n ω + 1 ’s hanging around on the right. ◮ Recall, we want to show R n = n ω / n ω + 1 is a constant, independent of ω ... Frame 46/121

Branching More with the happy-making thing Networks Introduction Finding Horton ratios: River Networks Definitions ◮ Letting Ω → ∞ , we have Allometry Laws Stream Ordering Horton’s Laws ∞ n ω T 1 ( R T ) k − 1 n ω + k � Tokunaga’s Law = 2 + (1) Horton ⇔ Tokunaga n ω + 1 n ω + 1 Reducing Horton Scaling relations k = 1 Fluctuations Models ◮ The ratio n ω + k / n ω + 1 can only be a function of k due References to self-similarity (which is implicit in Tokunaga’s law). ◮ The ratio n ω / n ω + 1 is independent of ω and depends only on T 1 and R T . ◮ Can now call n ω / n ω + 1 = R n . ◮ Immediately have n ω + k / n ω + 1 = R − ( k − 1 ) . n ◮ Plug into Eq. (1)... Frame 47/121

Branching More with the happy-making thing Networks Finding Horton ratios: Introduction River Networks ◮ Now have: Definitions Allometry ∞ Laws � Stream Ordering T 1 ( R T ) k − 1 R − ( k − 1 ) R n = 2 + Horton’s Laws n Tokunaga’s Law k = 1 Horton ⇔ Tokunaga Reducing Horton ∞ Scaling relations � ( R T / R n ) k − 1 = 2 + T 1 Fluctuations Models k = 1 References 1 = 2 + T 1 1 − R T / R n ◮ Rearrange to find: ( R n − 2 )( 1 − R T / R n ) = T 1 Frame 48/121

Branching More with the happy-making thing Networks Introduction Finding R n in terms of T 1 and R T : River Networks Definitions Allometry ◮ We are here: ( R n − 2 )( 1 − R T / R n ) = T 1 Laws Stream Ordering ◮ × R n to find quadratic in R n : Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ( R n − 2 )( R n − R T ) = T 1 R n Scaling relations Fluctuations Models References ◮ R 2 n − ( 2 + R T + T 1 ) R n + 2 R T = 0 ◮ Solution: � ( 2 + R T + T 1 ) 2 − 8 R T R n = ( 2 + R T + T 1 ) ± 2 Frame 49/121

Branching Finding other Horton ratios Networks Connect Tokunaga to R s Introduction River Networks ◮ Now use uniform drainage density ρ dd . Definitions Allometry Laws ◮ Assume side streams are roughly separated by Stream Ordering Horton’s Laws distance 1 /ρ dd . Tokunaga’s Law Horton ⇔ Tokunaga ◮ For an order ω stream segment, expected length is Reducing Horton Scaling relations Fluctuations � � Models ω − 1 � References s ω ≃ ρ − 1 ¯ 1 + T k dd k = 1 ◮ Substitute in Tokunaga’s law T k = T 1 R k − 1 : T � � ω − 1 � s ω ≃ ρ − 1 R k − 1 ∝ R ω ¯ 1 + T 1 T dd T k = 1 Frame 50/121

Branching Horton and Tokunaga are happy Networks Introduction Altogether then: River Networks Definitions Allometry ◮ Laws Stream Ordering ⇒ ¯ s ω / ¯ s ω − 1 = R T ⇒ R s = R T Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ Recall R ℓ = R s so Fluctuations Models References R ℓ = R T ◮ And from before: � ( 2 + R T + T 1 ) 2 − 8 R T R n = ( 2 + R T + T 1 ) + 2 Frame 51/121

Branching Horton and Tokunaga are happy Networks Introduction River Networks Definitions Allometry Some observations: Laws Stream Ordering Horton’s Laws ◮ R n and R ℓ depend on T 1 and R T . Tokunaga’s Law Horton ⇔ Tokunaga ◮ Seems that R a must as well... Reducing Horton Scaling relations ◮ Suggests Horton’s laws must contain some Fluctuations Models redundancy References ◮ We’ll in fact see that R a = R n . ◮ Also: Both Tokunaga’s law and Horton’s laws can be generalized to relationships between statistical distributions. [3, 4] Frame 52/121

Branching Horton and Tokunaga are happy Networks Introduction River Networks The other way round Definitions Allometry Laws ◮ Note: We can invert the expresssions for R n and R ℓ Stream Ordering Horton’s Laws to find Tokunaga’s parameters in terms of Horton’s Tokunaga’s Law Horton ⇔ Tokunaga parameters. Reducing Horton Scaling relations Fluctuations ◮ Models R T = R ℓ , References ◮ T 1 = R n − R ℓ − 2 + 2 R ℓ / R n . ◮ Suggests we should be able to argue that Horton’s laws imply Tokunaga’s laws (if drainage density is uniform)... Frame 53/121

(a) Branching Horton and Tokunaga are friends Networks ( � R ) l Introduction River Networks (b) From Horton to Tokunaga [2] Definitions Allometry Laws Stream Ordering ◮ Assume Horton’s laws Horton’s Laws Tokunaga’s Law hold for number and Horton ⇔ Tokunaga Reducing Horton length (c) Scaling relations Fluctuations ◮ Start with an order ω Models References stream ◮ Scale up by a factor of R ℓ , orders increment ◮ Maintain drainage density by adding new order 1 streams Frame 54/121

Branching Horton and Tokunaga are friends Networks Introduction . . . and in detail: River Networks Definitions Allometry ◮ Must retain same drainage density. Laws Stream Ordering ◮ Add an extra ( R ℓ − 1 ) first order streams for each Horton’s Laws Tokunaga’s Law original tributary. Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ Since number of first order streams is now given by Fluctuations Models T k + 1 we have: References � k � � T k + 1 = ( R ℓ − 1 ) T i + 1 . i = 1 ◮ For large ω , Tokunaga’s law is the solution—let’s check... Frame 55/121

Branching Horton and Tokunaga are friends Networks Just checking: Introduction River Networks ◮ Substitute Tokunaga’s law T i = T 1 R i − 1 = T 1 R i − 1 Definitions T ℓ Allometry Laws into � k Stream Ordering � Horton’s Laws � Tokunaga’s Law T k + 1 = ( R ℓ − 1 ) T i + 1 Horton ⇔ Tokunaga Reducing Horton i = 1 Scaling relations Fluctuations ◮ � k Models � References � T 1 R i − 1 T k + 1 = ( R ℓ − 1 ) + 1 ℓ i = 1 � R k � ℓ − 1 = ( R ℓ − 1 ) T 1 R ℓ − 1 + 1 R k R ℓ − 1 = T 1 R k ℓ ≃ ( R ℓ − 1 ) T 1 ... yep. ℓ Frame 56/121

Branching Horton’s laws of area and number: Networks The Mississippi The Mississippi 7 Introduction 0 6 −1 River Networks 5 (a) (b) −2 [source=/data6/dodds/work/rivers/dems/mississippi/figures/fignflipaomega_mispi10.ps] [source=/data6/dodds/work/rivers/dems/mississippi/figures/fignalomega_mispi10.ps] Definitions 4 Allometry −3 3 Laws −4 [15−Sep−2000 peter dodds] [15−Sep−2000 peter dodds] Stream Ordering 2 −5 Horton’s Laws 1 Tokunaga’s Law −6 Ω = 11 0 Horton ⇔ Tokunaga −7 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 Reducing Horton ω ω Scaling relations The Nile The Nile Fluctuations 7 7 10 10 n ω Models n Ω − ω +1 6 6 a ω (sq km) 10 10 a ω (sq km) References l ω (km) 5 5 10 10 [source=/data11/dodds/work/rivers/dems/HYDRO1K/africa/nile/figures/fignflipaomega_nile.ps] [source=/data11/dodds/work/rivers/dems/HYDRO1K/africa/nile/figures/fignalomega_nile.ps] 4 4 10 10 3 3 10 10 2 2 10 [10−Dec−1999 peter dodds] 10 [10−Dec−1999 peter dodds] 1 1 10 10 0 0 10 10 Ω = 10 −1 −1 10 10 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 stream order ω stream order ω ◮ In right plots, stream number graph has been flipped vertically. Frame 58/121 ◮ Highly suggestive that R n ≡ R a ...

Branching Measuring Horton ratios is tricky: Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga ◮ How robust are our estimates of ratios? Reducing Horton Scaling relations Fluctuations ◮ Rule of thumb: discard data for two smallest and two Models largest orders. References Frame 59/121

Branching Mississippi: Networks Introduction ω range R n R a R ℓ R s R a / R n River Networks [ 2 , 3 ] 5.27 5.26 2.48 2.30 1.00 Definitions Allometry [ 2 , 5 ] 4.86 4.96 2.42 2.31 1.02 Laws Stream Ordering [ 2 , 7 ] 4.77 4.88 2.40 2.31 1.02 Horton’s Laws Tokunaga’s Law [ 3 , 4 ] 4.72 4.91 2.41 2.34 1.04 Horton ⇔ Tokunaga Reducing Horton [ 3 , 6 ] 4.70 4.83 2.40 2.35 1.03 Scaling relations Fluctuations [ 3 , 8 ] 4.60 4.79 2.38 2.34 1.04 Models [ 4 , 6 ] 4.69 4.81 2.40 2.36 1.02 References [ 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 1.04 std dev σ 0.21 0.13 0.04 0.07 0.03 σ/µ 0.045 0.027 0.015 0.031 0.024 Frame 60/121

Branching Amazon: Networks Introduction River Networks ω range R n R a R ℓ R s R a / R n Definitions Allometry [ 2 , 3 ] 4.78 4.71 2.47 2.08 0.99 Laws Stream Ordering [ 2 , 5 ] 4.55 4.58 2.32 2.12 1.01 Horton’s Laws Tokunaga’s Law [ 2 , 7 ] 4.42 4.53 2.24 2.10 1.02 Horton ⇔ Tokunaga Reducing Horton [ 3 , 5 ] 4.45 4.52 2.26 2.14 1.01 Scaling relations Fluctuations [ 3 , 7 ] 4.35 4.49 2.20 2.10 1.03 Models [ 4 , 6 ] 4.38 4.54 2.22 2.18 1.03 References [ 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 1.02 std dev σ 0.17 0.10 0.10 0.09 0.02 σ/µ 0.038 0.023 0.045 0.042 0.019 Frame 61/121

Branching Reducing Horton’s laws: Networks Introduction Rough first effort to show R n ≡ R a : River Networks Definitions ◮ a Ω ∝ sum of all stream lengths in a order Ω basin Allometry Laws (assuming uniform drainage density) Stream Ordering Horton’s Laws Tokunaga’s Law ◮ So: Horton ⇔ Tokunaga Ω Reducing Horton � Scaling relations n ω ¯ a Ω ≃ s ω /ρ dd Fluctuations Models ω = 1 References n Ω Ω � ���� R Ω − ω s 1 · R ω − 1 ¯ ∝ · 1 n s � �� � � �� � ω = 1 n ω ¯ s ω Ω � R s � ω = R Ω � n ¯ s 1 R s R n ω = 1 Frame 62/121

Branching Reducing Horton’s laws: Networks Continued ... Introduction River Networks ◮ Definitions Allometry Ω � R s � ω a Ω ∝ R Ω � Laws n ¯ s 1 Stream Ordering R s R n Horton’s Laws Tokunaga’s Law ω = 1 Horton ⇔ Tokunaga Reducing Horton = R Ω 1 − ( R s / R n ) Ω R s Scaling relations n ¯ s 1 Fluctuations R s R n 1 − ( R s / R n ) Models References 1 ∼ R Ω − 1 ¯ s 1 1 − ( R s / R n ) as Ω ր n ◮ So, a Ω is growing like R Ω and therefore: n R n ≡ R a Frame 63/121

Branching Reducing Horton’s laws: Networks Introduction River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Not quite: Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ ... But this only a rough argument as Horton’s laws Scaling relations Fluctuations do not imply a strict hierarchy Models References ◮ Need to account for sidebranching. ◮ Problem set 1 question.... Frame 64/121

Branching Equipartitioning: Networks Introduction River Networks Intriguing division of area: Definitions Allometry ◮ Observe: Combined area of basins of order ω Laws Stream Ordering Horton’s Laws independent of ω . Tokunaga’s Law Horton ⇔ Tokunaga ◮ Not obvious: basins of low orders not necessarily Reducing Horton Scaling relations contained in basis on higher orders. Fluctuations Models ◮ Story: References R n ≡ R a ⇒ n ω ¯ a ω = const ◮ Reason: n ω ∝ ( R n ) − ω a ω ∝ ( R a ) ω ∝ n − 1 ¯ ω Frame 65/121

Branching Equipartitioning: Networks Some examples: Introduction River Networks Mississippi basin partitioning Amazon basin partitioning Definitions 1 1 Allometry Laws 0.8 0.8 Stream Ordering Horton’s Laws Tokunaga’s Law n ω a ω / a Ω n ω a ω / a Ω [source=/data6/dodds/work/rivers/dems/mississippi/figures/figequipart_mispi.ps] 0.6 0.6 [source=/data6/dodds/work/rivers/dems/amazon/figures/figequipart_amazon.ps] Horton ⇔ Tokunaga Reducing Horton 0.4 0.4 Scaling relations [15−Dec−2000 peter dodds] [15−Dec−2000 peter dodds] Fluctuations Models 0.2 0.2 References 0 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 ω ω Nile basin partitioning 1 0.8 [source=/data11/dodds/work/rivers/dems/HYDRO1K/africa/nile/figures/figequipart_nile.ps] n ω a ω / a Ω 0.6 0.4 [15−Dec−2000 peter dodds] 0.2 Frame 66/121 0 1 2 3 4 5 6 7 8 9 10 ω

Branching Scaling laws Networks Introduction River Networks Definitions The story so far: Allometry Laws Stream Ordering ◮ Natural branching networks are hierarchical, Horton’s Laws Tokunaga’s Law self-similar structures Horton ⇔ Tokunaga Reducing Horton ◮ Hierarchy is mixed Scaling relations Fluctuations ◮ Tokunaga’s law describes detailed architecture: Models References T k = T 1 R k − 1 . T ◮ We have connected Tokunaga’s and Horton’s laws ◮ Only two Horton laws are independent ( R n = R a ) ◮ Only two parameters are independent: ( T 1 , R T ) ⇔ ( R n , R s ) Frame 68/121

Branching Scaling laws Networks Introduction River Networks Definitions A little further... Allometry Laws ◮ Ignore stream ordering for the moment Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Pick a random location on a branching network p . Horton ⇔ Tokunaga Reducing Horton ◮ Each point p is associated with a basin and a longest Scaling relations Fluctuations stream length Models References ◮ Q: What is probability that the p ’s drainage basin has area a ? P ( a ) ∝ a − τ for large a ◮ Q: What is probability that the longest stream from p has length ℓ ? P ( ℓ ) ∝ ℓ − γ for large ℓ ◮ Roughly observed: 1 . 3 � τ � 1 . 5 and 1 . 7 � γ � 2 . 0 Frame 69/121

Branching Scaling laws Networks Introduction River Networks Definitions Probability distributions with power-law decays Allometry Laws ◮ We see them everywhere: Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Earthquake magnitudes (Gutenberg-Richter law) Horton ⇔ Tokunaga ◮ City sizes (Zipf’s law) Reducing Horton Scaling relations ◮ Word frequency (Zipf’s law) [24] Fluctuations Models ◮ Wealth (maybe not—at least heavy tailed) References ◮ Statistical mechanics (phase transitions) [5] ◮ A big part of the story of complex systems ◮ Arise from mechanisms: growth, randomness, optimization, ... ◮ Our task is always to illuminate the mechanism... Frame 70/121

Branching Scaling laws Networks Introduction River Networks Connecting exponents Definitions Allometry Laws ◮ We have the detailed picture of branching networks Stream Ordering Horton’s Laws (Tokunaga and Horton) Tokunaga’s Law Horton ⇔ Tokunaga ◮ Plan: Derive P ( a ) ∝ a − τ and P ( ℓ ) ∝ ℓ − γ starting with Reducing Horton Scaling relations Tokunaga/Horton story [20, 1, 2] Fluctuations Models ◮ Let’s work on P ( ℓ ) ... References ◮ Our first fudge: assume Horton’s laws hold throughout a basin of order Ω . ◮ (We know they deviate from strict laws for low ω and high ω but not too much.) ◮ Next: place stick between teeth. Bite stick. Proceed. Frame 71/121

Branching Scaling laws Networks Introduction River Networks Finding γ : Definitions Allometry Laws ◮ Often useful to work with cumulative distributions, Stream Ordering Horton’s Laws especially when dealing with power-law distributions. Tokunaga’s Law Horton ⇔ Tokunaga ◮ The complementary cumulative distribution turns out Reducing Horton Scaling relations Fluctuations to be most useful: Models References � ℓ max P > ( ℓ ∗ ) = P ( ℓ > ℓ ∗ ) = P ( ℓ ) d ℓ ℓ = ℓ ∗ ◮ P > ( ℓ ∗ ) = 1 − P ( ℓ < ℓ ∗ ) ◮ Also known as the exceedance probability. Frame 72/121

Branching Scaling laws Networks Finding γ : Introduction River Networks ◮ The connection between P ( x ) and P > ( x ) when P ( x ) Definitions Allometry has a power law tail is simple: Laws Stream Ordering ◮ Given P ( ℓ ) ∼ ℓ − γ large ℓ then for large enough ℓ ∗ Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga � ℓ max Reducing Horton Scaling relations P > ( ℓ ∗ ) = P ( ℓ ) d ℓ Fluctuations Models ℓ = ℓ ∗ References � ℓ max ℓ − γ d ℓ ∼ ℓ = ℓ ∗ � ℓ max ℓ − γ + 1 � = � − γ + 1 � ℓ = ℓ ∗ ∝ ℓ − γ + 1 for ℓ max ≫ ℓ ∗ ∗ Frame 73/121

Branching Scaling laws Networks Introduction Finding γ : River Networks Definitions ◮ Aim: determine probability of randomly choosing a Allometry Laws point on a network with main stream length > ℓ ∗ Stream Ordering Horton’s Laws ◮ Assume some spatial sampling resolution ∆ Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Landscape is broken up into grid of ∆ × ∆ sites Scaling relations Fluctuations ◮ Approximate P > ( ℓ ∗ ) as Models References P > ( ℓ ∗ ) = N > ( ℓ ∗ ; ∆) N > ( 0 ; ∆) . where N > ( ℓ ∗ ; ∆) is the number of sites with main stream length > ℓ ∗ . ◮ Use Horton’s law of stream segments: s ω / s ω − 1 = R s ... Frame 74/121

Branching Scaling laws Networks Introduction Finding γ : River Networks Definitions ◮ Set ℓ ∗ = ℓ ω for some 1 ≪ ω ≪ Ω . Allometry Laws Stream Ordering ◮ Horton’s Laws � Ω ω ′ = ω + 1 n ω ′ s ω ′ / ∆ Tokunaga’s Law P > ( ℓ ω ) = N > ( ℓ ω ; ∆) Horton ⇔ Tokunaga N > ( 0 ; ∆) ≃ Reducing Horton � Ω ω ′ = 1 n ω ′ s ω ′ / ∆ Scaling relations Fluctuations Models ◮ ∆ ’s cancel References ◮ Denominator is a Ω ρ dd , a constant. ◮ So... using Horton’s laws... Ω Ω � � ( 1 · R Ω − ω ′ s 1 · R ω ′ − 1 n ω ′ s ω ′ ≃ )(¯ P > ( ℓ ω ) ∝ ) n s ω ′ = ω + 1 ω ′ = ω + 1 Frame 75/121

Branching Scaling laws Networks Finding γ : Introduction River Networks ◮ We are here: Definitions Allometry Laws Stream Ordering Ω � Horton’s Laws ( 1 · R Ω − ω ′ s 1 · R ω ′ − 1 )(¯ P > ( ℓ ω ) ∝ ) Tokunaga’s Law n s Horton ⇔ Tokunaga Reducing Horton ω ′ = ω + 1 Scaling relations Fluctuations ◮ Cleaning up irrelevant constants: Models References Ω � R s � ω ′ � P > ( ℓ ω ) ∝ R n ω ′ = ω + 1 ◮ Change summation order by substituting ω ′′ = Ω − ω ′ . ◮ Sum is now from ω ′′ = 0 to ω ′′ = Ω − ω − 1 (equivalent to ω ′ = Ω down to ω ′ = ω + 1) Frame 76/121

Branching Scaling laws Networks Introduction River Networks Finding γ : Definitions Allometry Laws ◮ Stream Ordering Horton’s Laws Tokunaga’s Law Ω − ω − 1 Ω − ω − 1 � R s � Ω − ω ′′ � R n � ω ′′ Horton ⇔ Tokunaga � � Reducing Horton P > ( ℓ ω ) ∝ ∝ Scaling relations R n R s Fluctuations ω ′′ = 0 ω ′′ = 0 Models References ◮ Since R n < R s and 1 ≪ ω ≪ Ω , � R n � Ω − ω � R n � − ω P > ( ℓ ω ) ∝ ∝ R s R s again using � n i = 0 a n = ( a i + 1 − 1 ) / ( a − 1 ) Frame 77/121

Branching Scaling laws Networks Introduction River Networks Finding γ : Definitions Allometry Laws ◮ Nearly there: Stream Ordering Horton’s Laws Tokunaga’s Law � R n � − ω Horton ⇔ Tokunaga = e − ω ln ( R n / R s ) Reducing Horton P > ( ℓ ω ) ∝ Scaling relations R s Fluctuations Models References ◮ Need to express right hand side in terms of ℓ ω . ◮ Recall that ℓ ω ≃ ¯ ℓ 1 R ω − 1 . ℓ ◮ ℓ ω ∝ R ω ℓ = R ω s = e ω ln R s Frame 78/121

Branching Scaling laws Networks Finding γ : Introduction River Networks ◮ Therefore: Definitions Allometry Laws � e ω ln R s � − ln ( R n / R s ) / ln ( R s ) P > ( ℓ ω ) ∝ e − ω ln ( R n / R s ) = Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations ◮ Models ∝ ℓ ω − ln ( R n / R s ) / ln R s References ◮ = ℓ − ( ln R n − ln R s ) / ln R s ω ◮ = ℓ − ln R n / ln R s + 1 ω ◮ = ℓ − γ + 1 ω Frame 79/121

Branching Scaling laws Networks Introduction River Networks Finding γ : Definitions Allometry Laws ◮ And so we have: Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga γ = ln R n / ln R s Reducing Horton Scaling relations Fluctuations Models ◮ Proceeding in a similar fashion, we can show References τ = 2 − ln R s / ln R n = 2 − 1 /γ ◮ Such connections between exponents are called scaling relations ◮ Let’s connect to one last relationship: Hack’s law Frame 80/121

Branching Scaling laws Networks Hack’s law: [6] Introduction River Networks ◮ Definitions Allometry ℓ ∝ a h Laws Stream Ordering Horton’s Laws ◮ Typically observed that 0 . 5 � h � 0 . 7. Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Use Horton laws to connect h to Horton ratios: Scaling relations Fluctuations Models ℓ ω ∝ R ω s and a ω ∝ R ω n References ◮ Observe: � e ω ln R n � ln R s / ln R n ℓ ω ∝ e ω ln R s ∝ n ) ln R s / ln R n = a ln R s / ln R n ∝ ( R ω ⇒ h = ln R s / ln R n ω Frame 81/121

Branching Connecting exponents Networks Only 3 parameters are independent: Introduction e.g., take d , R n , and R s River Networks Definitions scaling relation/parameter: [2] Allometry relation: Laws ℓ ∼ L d Stream Ordering d Horton’s Laws T k = T 1 ( R T ) k − 1 Tokunaga’s Law T 1 = R n − R s − 2 + 2 R s / R n Horton ⇔ Tokunaga Reducing Horton R T = R s Scaling relations Fluctuations n ω / n ω + 1 = R n R n Models ¯ a ω + 1 / ¯ a ω = R a R a = R n References ℓ ω + 1 / ¯ ¯ ℓ ω = R ℓ R ℓ = R s ℓ ∼ a h h = log R s / log R n a ∼ L D D = d / h L ⊥ ∼ L H H = d / h − 1 P ( a ) ∼ a − τ τ = 2 − h P ( ℓ ) ∼ ℓ − γ γ = 1 / h Λ ∼ a β β = 1 + h λ ∼ L ϕ Frame 82/121 ϕ = d

Branching Equipartitioning reexamined: Networks Recall this story: Introduction River Networks Mississippi basin partitioning Amazon basin partitioning Definitions 1 1 Allometry Laws 0.8 0.8 Stream Ordering Horton’s Laws Tokunaga’s Law n ω a ω / a Ω n ω a ω / a Ω [source=/data6/dodds/work/rivers/dems/mississippi/figures/figequipart_mispi.ps] 0.6 0.6 [source=/data6/dodds/work/rivers/dems/amazon/figures/figequipart_amazon.ps] Horton ⇔ Tokunaga Reducing Horton 0.4 0.4 Scaling relations [15−Dec−2000 peter dodds] [15−Dec−2000 peter dodds] Fluctuations Models 0.2 0.2 References 0 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 ω ω Nile basin partitioning 1 0.8 [source=/data11/dodds/work/rivers/dems/HYDRO1K/africa/nile/figures/figequipart_nile.ps] n ω a ω / a Ω 0.6 0.4 [15−Dec−2000 peter dodds] 0.2 Frame 83/121 0 1 2 3 4 5 6 7 8 9 10 ω

Branching Equipartitioning Networks Introduction River Networks Definitions Allometry Laws ◮ What about Stream Ordering Horton’s Laws P ( a ) ∼ a − τ ? Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Since τ > 1, suggests no equipartitioning: Scaling relations Fluctuations Models aP ( a ) ∼ a − τ + 1 � = const References ◮ P ( a ) overcounts basins within basins... ◮ while stream ordering separates basins... Frame 84/121

Branching Fluctuations Networks Introduction River Networks Moving beyond the mean: Definitions Allometry Laws Stream Ordering ◮ Both Horton’s laws and Tokunaga’s law relate Horton’s Laws Tokunaga’s Law average properties, e.g., Horton ⇔ Tokunaga Reducing Horton Scaling relations s ω / ¯ ¯ s ω − 1 = R s Fluctuations Models References ◮ Natural generalization to consideration relationships between probability distributions ◮ Yields rich and full description of branching network structure ◮ See into the heart of randomness... Frame 86/121

Branching A toy model—Scheidegger’s model Networks Introduction Directed random networks [12, 13] River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations Models References ◮ ◮ P ( ց ) = P ( ւ ) = 1 / 2 ◮ Flow is directed downwards ◮ Useful and interesting test case—more later... Frame 87/121

Branching Generalizing Horton’s laws Networks Introduction ℓ ω ∝ ( R ℓ ) ω ⇒ N ( ℓ | ω ) = ( R n R ℓ ) − ω F ℓ ( ℓ/ R ω ◮ ¯ ℓ ) River Networks a ω ∝ ( R a ) ω ⇒ N ( a | ω ) = ( R 2 n ) − ω F a ( a / R ω ◮ ¯ Definitions n ) Allometry Laws Stream Ordering Mississippi: length distributions Mississippi: length distributions Horton’s Laws 2 −3 10 10 ω =3 Tokunaga’s Law R n = 4.69, R l = 2.38 Horton ⇔ Tokunaga 4 Reducing Horton 5 −4 Scaling relations 10 6 Fluctuations 0 ω N( l | ω ) 10 [source=/data6/dodds/work/rivers/dems/mississippi/figures/figlw_collapse_mispi2a.ps] [source=/data6/dodds/work/rivers/dems/mississippi/figures/figlw_collapse_mispi2.ps] Models N( l | ω ) References −5 10 ω − Ω R l [09−Dec−1999 peter dodds] −2 [09−Dec−1999 peter dodds] R n 10 −6 ω =3 10 4 5 6 −7 −4 10 10 0 1 2 3 0 100 200 300 400 − ω l R l l (km) ◮ Scaling collapse works well for intermediate orders ◮ All moments grow exponentially with order Frame 88/121

Branching Generalizing Horton’s laws Networks Introduction River Networks Definitions ◮ How well does overall basin fit internal pattern? Allometry Laws Stream Ordering Mississippi −7 Horton’s Laws 1.4x 10 ◮ Actual length = 4920 km Tokunaga’s Law ω =4 Horton ⇔ Tokunaga (at 1 km res) ω =3 1.2 Reducing Horton actual l Scaling relations < l > ◮ Predicted Mean length Ω − ω ) Fluctuations 1 Models − Ω + ω P( l R l [source=/data6/dodds/work/rivers/dems/mississippi/figures/figlw_blownup.ps] = 11100 km 0.8 References ◮ Predicted Std dev = 0.6 [10−Dec−1999 peter dodds] 5600 km R l 0.4 0.2 ◮ Actual length/Mean 0 length = 44 % 0 1 2 3 4 7 Ω − ω (m) x 10 l R l ◮ Okay. Frame 89/121

Branching Generalizing Horton’s laws Networks Introduction Comparison of predicted versus measured main stream lengths for large scale river networks (in 10 3 km): River Networks Definitions Allometry ¯ ℓ/ ¯ σ ℓ / ¯ Laws basin: ℓ Ω ℓ Ω σ ℓ ℓ Ω ℓ Ω Stream Ordering Horton’s Laws Mississippi 4.92 11.10 5.60 0.44 0.51 Tokunaga’s Law Horton ⇔ Tokunaga Amazon 5.75 9.18 6.85 0.63 0.75 Reducing Horton Scaling relations Nile 6.49 2.66 2.20 2.44 0.83 Fluctuations Models Congo 5.07 10.13 5.75 0.50 0.57 References Kansas 1.07 2.37 1.74 0.45 0.73 ¯ a / ¯ σ a / ¯ a a Ω σ a a Ω a Ω Mississippi 2.74 7.55 5.58 0.36 0.74 Amazon 5.40 9.07 8.04 0.60 0.89 Nile 3.08 0.96 0.79 3.19 0.82 Congo 3.70 10.09 8.28 0.37 0.82 Kansas 0.14 0.49 0.42 0.28 0.86 Frame 90/121

Branching Combining stream segments distributions: Networks Introduction River Networks Definitions ◮ Stream segments Allometry Laws sum to give main Stream Ordering Horton’s Laws stream lengths Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton ◮ Scaling relations Fluctuations µ = ω Models � References ℓ ω = s µ µ = 1 ◮ P ( ℓ ω ) is a convolution of distributions for the s ω Frame 91/121

Branching Generalizing Horton’s laws Networks Introduction ◮ Sum of variables ℓ ω = � µ = ω River Networks µ = 1 s µ leads to convolution Definitions of distributions: Allometry Laws Stream Ordering Horton’s Laws N ( ℓ | ω ) = N ( s | 1 ) ∗ N ( s | 2 ) ∗ · · · ∗ N ( s | ω ) Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations Models Mississippi: stream segments −3 References R n = 4.69, R l = 2.33 (b) ω , ω ) 1 ω P( l (s) −4 F ( s / R ω N ( s | ω ) = ℓ ) [source=/data6/dodds/work/rivers/dems/mississippi/figures/figellw_collapse_mispi2a.ps] R ω n R ω (s) ℓ ω − Ω R l −5 [07−Dec−2000 peter dodds] log 10 R n F ( x ) = e − x /ξ −6 Mississippi: ξ ≃ 900 m. −7 0 1 2 3 − ω l (s) ω R l (s) Frame 92/121

Branching Generalizing Horton’s laws Networks Introduction ◮ Next level up: Main stream length distributions must River Networks Definitions combine to give overall distribution for stream length Allometry Laws Stream Ordering Mississippi: length distributions Horton’s Laws 2 10 Tokunaga’s Law ω =3 Horton ⇔ Tokunaga Reducing Horton 4 Scaling relations 5 Fluctuations 6 Models 0 3−6 [source=/data6/dodds/work/rivers/dems/mississippi/figures/figlw_powerlawsum_mispi.ps] 10 ◮ P ( ℓ ) ∼ ℓ − γ References N( l | ω ) ◮ Another round of convolutions [3] [22−Mar−2000 peter dodds] −2 10 ◮ Interesting... −4 10 1 2 10 10 l (km) Frame 93/121

Branching Generalizing Horton’s laws Networks Introduction −4 River Networks 1.2x 10 Definitions Allometry −4 Laws x 10 Stream Ordering 1 Horton’s Laws 4 Tokunaga’s Law Horton ⇔ Tokunaga 3 0.8 Reducing Horton Number and area Scaling relations 2 Fluctuations distributions for the Models 0.6 1 References Scheidegger model 0 P ( n 1 , 6 ) versus P ( a 6 ) . 0.4 0 1 2 3 4 x 10 0.2 0 0 1 2 3 4 4 x 10 Frame 94/121

Branching Generalizing Tokunaga’s law Networks Introduction Scheidegger: River Networks Definitions −1 Allometry 1 (a) Laws (b) Stream Ordering Horton’s Laws log 10 (R l (s) ) µ P(T µ , ν ) −2 Tokunaga’s Law 0 log 10 P(T µ , ν ) Horton ⇔ Tokunaga Reducing Horton Scaling relations −3 −1 Fluctuations Models References −4 −2 −5 −3 0 100 200 300 0 0.1 0.2 0.3 0.4 0.5 0.6 T µ , ν (R l (s) ) − µ T µ , ν ◮ Observe exponential distributions for T µ,ν ◮ Scaling collapse works using R s Frame 95/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Mississippi: Definitions Allometry 2.5 Laws 3.5 (a) Stream Ordering (b) Horton’s Laws 2 3 Tokunaga’s Law (s) ) − ν P(T µ , ν ) Horton ⇔ Tokunaga log 10 P(T µ , ν ) Reducing Horton 1.5 2.5 Scaling relations Fluctuations Models 2 1 References log 10 (R l 1.5 0.5 1 0 0 20 40 60 0.5 0 1 2 3 4 5 T µ , ν (R l (s) ) ν T µ , ν ◮ Same data collapse for Mississippi... Frame 96/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions So Allometry � T µ,ν / ( R s ) µ − ν − 1 � Laws P ( T µ,ν ) = ( R s ) µ − ν − 1 P t Stream Ordering Horton’s Laws Tokunaga’s Law where Horton ⇔ Tokunaga Reducing Horton P t ( z ) = 1 e − z /ξ t . Scaling relations Fluctuations ξ t Models References P ( s µ ) ⇔ P ( T µ,ν ) ◮ Exponentials arise from randomness. ◮ Look at joint probability P ( s µ , T µ,ν ) . Frame 97/121

Branching Generalizing Tokunaga’s law Networks Introduction � River Networks � � 1 Definitions Network architecture: Allometry � � 2 Laws Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Inter-tributary Horton ⇔ Tokunaga Reducing Horton lengths Scaling relations Fluctuations exponentially Models distributed References ◮ Leads to random spatial distribution of stream segments Frame 98/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions Allometry ◮ Follow streams segments down stream from their Laws Stream Ordering beginning Horton’s Laws Tokunaga’s Law ◮ Probability (or rate) of an order µ stream segment Horton ⇔ Tokunaga Reducing Horton terminating is constant: Scaling relations Fluctuations Models p µ ≃ 1 / ( R s ) µ − 1 ξ s ˜ References ◮ Probability decays exponentially with stream order ◮ Inter-tributary lengths exponentially distributed ◮ ⇒ random spatial distribution of stream segments Frame 99/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks ◮ Joint distribution for generalized version of Definitions Allometry Tokunaga’s law: Laws Stream Ordering Horton’s Laws � s µ − 1 � Tokunaga’s Law p T µ,ν p µ ) s µ − T µ,ν − 1 P ( s µ , T µ,ν ) = ˜ ( 1 − p ν − ˜ p µ Horton ⇔ Tokunaga ν Reducing Horton T µ,ν Scaling relations Fluctuations Models where References ◮ p ν = probability of absorbing an order ν side stream ◮ ˜ p µ = probability of an order µ stream terminating ◮ Approximation: depends on distance units of s µ ◮ In each unit of distance along stream, there is one chance of a side stream entering or the stream terminating. Frame 100/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks ◮ Now deal with thing: Definitions Allometry Laws � s µ − 1 � Stream Ordering p T µ,ν p µ ) s µ − T µ,ν − 1 P ( s µ , T µ,ν ) = ˜ ( 1 − p ν − ˜ p µ Horton’s Laws ν T µ,ν Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations ◮ Set ( x , y ) = ( s µ , T µ,ν ) and q = 1 − p ν − ˜ p µ , Fluctuations Models approximate liberally. References ◮ Obtain P ( x , y ) = Nx − 1 / 2 [ F ( y / x )] x where � 1 − v � − ( 1 − v ) � v � − v F ( v ) = . q p Frame 101/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions ◮ Checking form of P ( s µ , T µ,ν ) works: Allometry Laws Stream Ordering Scheidegger: Horton’s Laws Tokunaga’s Law 1 1.5 Horton ⇔ Tokunaga (a) (b) Reducing Horton Scaling relations 0.8 1 Fluctuations Models 0.5 0.6 (s) (s) ) References [F(v)] l µ P(v | l µ 0 0.4 −0.5 0.2 −1 0 −1.5 0 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 (s) (s) v = T µ , ν / l µ v = T µ , ν / l µ Frame 102/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions ◮ Checking form of P ( s µ , T µ,ν ) works: Allometry Laws Stream Ordering Scheidegger: Horton’s Laws Tokunaga’s Law 1.5 Horton ⇔ Tokunaga −0.5 (a) (b) Reducing Horton Scaling relations 1 −1 Fluctuations (s) ) (s) / T µ , ν ) Models −1.5 log 10 P(T µ , ν / l µ 0.5 References −2 0 log 10 P(l µ −2.5 −0.5 −3 −1 −3.5 −1.5 −4 0 0.1 0.2 0.3 0 10 20 30 40 50 (s) (s) / T µ , ν T µ , ν / l µ l µ Frame 103/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions ◮ Checking form of P ( s µ , T µ,ν ) works: Allometry Laws Stream Ordering Scheidegger: Horton’s Laws Tokunaga’s Law (s) ) Horton ⇔ Tokunaga 1 (b) Reducing Horton 2 (a) log 10 (R l (s) ) − µ /2+ ν /2 P(T µ , ν / l µ Scaling relations Fluctuations 0.5 (s) ) 1.5 Models log 10 P(T µ , ν / l µ References 0 1 −0.5 0.5 −1 0 −1.5 −0.5 −0.2 −0.1 0 0.1 0.2 0 0.05 0.1 0.15 (s) − ρ ν ] (R l (s) ) µ /2− ν /2 (s) [T µ , ν / l µ T µ , ν / l µ Frame 104/121

Branching Generalizing Tokunaga’s law Networks Introduction River Networks Definitions ◮ Checking form of P ( s µ , T µ,ν ) works: Allometry Laws Stream Ordering Mississippi: Horton’s Laws Tokunaga’s Law 1.5 Horton ⇔ Tokunaga 0.8 (a) Reducing Horton (b) (s) ) Scaling relations (s) ) − ν P(T µ , ν / l µ Fluctuations (s) ) 1 0.4 Models log 10 P(T µ , ν / l µ References 0 0.5 log 10 (R l −0.4 0 −0.8 0 0.15 0.3 0.45 0.6 −0.5 −0.25 0 0.25 0.5 (s) ) ν (s) (s) − ρ ν ](R l T µ , ν / l µ [T µ , ν / l µ Frame 105/121

Branching Models Networks Introduction Random subnetworks on a Bethe lattice [15] River Networks ◮ Dominant theoretical concept Definitions Allometry Laws for several decades. Stream Ordering Horton’s Laws ◮ Bethe lattices are fun and Tokunaga’s Law Horton ⇔ Tokunaga tractable. Reducing Horton Scaling relations ◮ Led to idea of “Statistical Fluctuations Models inevitability” of river network References statistics [8] ◮ But Bethe lattices unconnected with surfaces. ◮ In fact, Bethe lattices ≃ infinite dimensional spaces (oops). ◮ So let’s move on... Frame 107/121

Branching Scheidegger’s model Networks Introduction Directed random networks [12, 13] River Networks Definitions Allometry Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations Models References ◮ ◮ P ( ց ) = P ( ւ ) = 1 / 2 ◮ Functional form of all scaling laws exhibited but exponents differ from real world [18, 19, 17] Frame 108/121

Branching A toy model—Scheidegger’s model Networks Introduction River Networks Definitions Allometry Laws Random walk basins: Stream Ordering Horton’s Laws Tokunaga’s Law ◮ Boundaries of basins are random walks Horton ⇔ Tokunaga Reducing Horton x Scaling relations Fluctuations Models References area a n Frame 109/121

Branching Scheidegger’s model Networks Introduction 9 9 Increasing partition of N=64 River Networks Definitions n Allometry 8 8 8 8 Laws Stream Ordering Horton’s Laws Tokunaga’s Law Horton ⇔ Tokunaga 6 6 Reducing Horton Scaling relations Fluctuations Models References 2 x Frame 110/121

Branching Scheidegger’s model Networks Introduction Prob for first return of a random walk in (1+1) River Networks dimensions: Definitions Allometry Laws Stream Ordering ◮ Horton’s Laws 1 2 √ π n − 3 / 2 . Tokunaga’s Law P ( n ) ∼ Horton ⇔ Tokunaga Reducing Horton Scaling relations Fluctuations and so P ( ℓ ) ∝ ℓ − 3 / 2 . Models References ◮ Typical area for a walk of length n is ∝ n 3 / 2 : ℓ ∝ a 2 / 3 . ◮ Find τ = 4 / 3, h = 2 / 3, γ = 3 / 2, d = 1. ◮ Note τ = 2 − h and γ = 1 / h . ◮ R n and R ℓ have not been derived analytically. Frame 111/121