Supply Networks Optimal Supply Networks Introduction Optimal branching Complex Networks, Course 295A, Spring, 2008 Murray meets Tokunaga Single Source History Reframing the question Minimal volume calculation Prof. Peter Dodds Blood networks River networks Distributed Department of Mathematics & Statistics Sources University of Vermont Facility location Size-density law Cartograms References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/85
Supply Networks Outline Introduction Introduction Optimal branching Optimal branching Murray meets Tokunaga Murray meets Tokunaga Single Source History Reframing the question Single Source Minimal volume calculation Blood networks History River networks Reframing the question Distributed Sources Minimal volume calculation Facility location Size-density law Blood networks Cartograms References River networks Distributed Sources Facility location Size-density law Cartograms References Frame 2/85
Supply Networks Optimal supply networks Introduction Optimal branching Murray meets Tokunaga Single Source What’s the best way to distribute stuff? History Reframing the question Minimal volume calculation ◮ Stuff = medical services, energy, people, Blood networks River networks ◮ Some fundamental network problems: Distributed Sources 1. Distribute stuff from a single source to many sinks Facility location Size-density law 2. Distribute stuff from many sources to many sinks Cartograms 3. Redistribute stuff between nodes that are both References sources and sinks ◮ Supply and Collection are equivalent problems Frame 3/85
Supply Networks River network models Introduction Optimal branching Murray meets Tokunaga Optimality: Single Source History Reframing the question ◮ Optimal channel networks [10] Minimal volume calculation Blood networks River networks ◮ Thermodynamic analogy [11] Distributed Sources Facility location versus... Size-density law Cartograms Randomness: References ◮ Scheidegger’s directed random networks ◮ Undirected random networks Frame 4/85
Supply Networks Optimization approaches Cardiovascular networks: Introduction Optimal branching ◮ Murray’s law (1926) connects branch radii at forks: [8] Murray meets Tokunaga Single Source History r 3 0 = r 3 1 + r 3 Reframing the question 2 Minimal volume calculation Blood networks River networks where r 0 = radius of main branch Distributed Sources and r 1 and r 2 are radii of sub-branches Facility location Size-density law ◮ Calculation assumes Poiseuille flow Cartograms References ◮ Holds up well for outer branchings of blood networks ◮ Also found to hold for trees ◮ Use hydraulic equivalent of Ohm’s law: ∆ p = Φ Z ⇔ V = IR where ∆ p = pressure difference, Φ = flux Frame 5/85
Supply Networks Optimization approaches Cardiovascular networks: Introduction Optimal branching ◮ Fluid mechanics: Poiseuille impedance for smooth Murray meets Tokunaga Single Source flow in a tube of radius r and length ℓ : History Reframing the question Minimal volume calculation Z = 8 ηℓ Blood networks River networks π r 4 Distributed Sources where η = dynamic viscosity Facility location Size-density law ◮ Power required to overcome impedance: Cartograms References P drag = Φ∆ p = Φ 2 Z ◮ Also have rate of energy expenditure in maintaining blood: P metabolic = cr 2 ℓ where c is a metabolic constant. Frame 6/85
Supply Networks Optimization approaches Introduction Optimal branching Murray meets Tokunaga Single Source Aside on P drag History Reframing the question Minimal volume calculation Blood networks ◮ Work done = F · d = energy transferred by force F River networks ◮ Power = rate work is done = F · v Distributed Sources Facility location ◮ ∆ P = Force per unit area Size-density law Cartograms ◮ Φ = Volume per unit time References = cross-sectional area · velocity ◮ So Φ∆ P = Force · velocity Frame 7/85
Supply Networks Optimization approaches Introduction Optimal branching Murray’s law: Murray meets Tokunaga Single Source ◮ Total power (cost): History Reframing the question Minimal volume calculation Blood networks P = P drag + P metabolic = Φ 2 8 ηℓ π r 4 + cr 2 ℓ River networks Distributed Sources Facility location Size-density law Cartograms ◮ Observe power increases linearly with ℓ References ◮ But r ’s effect is nonlinear: ◮ increasing r makes flow easier but increases metabolic cost (as r 2 ) ◮ decreasing r decrease metabolic cost but impedance goes up (as r − 4 ) Frame 8/85
Supply Networks Optimization Introduction Murray’s law: Optimal branching Murray meets Tokunaga ◮ Minimize P with respect to r : Single Source History � � Reframing the question ∂ P ∂ r = ∂ Φ 2 8 ηℓ Minimal volume calculation π r 4 + cr 2 ℓ Blood networks ∂ r River networks Distributed Sources = − 4 Φ 2 8 ηℓ Facility location π r 5 + c 2 r ℓ = 0 Size-density law Cartograms References ◮ Rearrange/cancel/slap: Φ 2 = c π r 6 16 η = k 2 r 6 where k = constant. Frame 9/85
Supply Networks Optimization Introduction Murray’s law: Optimal branching Murray meets Tokunaga ◮ So we now have: Single Source Φ = kr 3 History Reframing the question Minimal volume calculation ◮ Flow rates at each branching have to add up (else Blood networks River networks our organism is in serious trouble...): Distributed Sources Facility location Φ 0 = Φ 1 + Φ 2 Size-density law Cartograms References where again 0 refers to the main branch and 1 and 2 refers to the offspring branches ◮ All of this means we have a groovy cube-law: r 3 0 = r 3 1 + r 3 2 Frame 10/85
Supply Networks Optimization Introduction Murray meets Tokunaga: Optimal branching Murray meets Tokunaga ◮ Φ ω = volume rate of flow into an order ω vessel Single Source segment History Reframing the question Minimal volume calculation ◮ Tokunaga picture: Blood networks River networks ω − 1 Distributed � Sources Φ ω = 2 Φ ω − 1 + T k Φ ω − k Facility location Size-density law k = 1 Cartograms References ◮ Using φ ω = kr 3 ω ω − 1 � r 3 ω = 2 r 3 T k r 3 ω − 1 + ω − k k = 1 ◮ Find Horton ratio for vessell radius R r = r ω / r ω − 1 ... Frame 12/85
Supply Networks Optimization Introduction Optimal branching Murray meets Tokunaga Single Source Murray meets Tokunaga: History Reframing the question Minimal volume calculation ◮ Find R 3 Blood networks r satisfies same equation as R n and R v River networks ( v is for volume): Distributed Sources Facility location R 3 r = R n = R v = R 3 Size-density law n Cartograms References ◮ Is there more we could do here to constrain the Horton ratios and Tokunaga constants? Frame 13/85
Supply Networks Optimization Introduction Optimal branching Murray meets Tokunaga Murray meets Tokunaga: Single Source History ◮ Isometry: V ω ∝ ℓ 3 Reframing the question ω Minimal volume calculation Blood networks ◮ Gives River networks R 3 Distributed ℓ = R v = R n Sources Facility location ◮ We need one more constraint... Size-density law Cartograms ◮ West et al (1997) [16] achieve similar results following References Horton’s laws. ◮ So does Turcotte et al. (1998) [15] using Tokunaga (sort of). Frame 14/85
Supply Networks Optimization approaches Introduction Optimal branching Murray meets Tokunaga The bigger picture: Single Source History Reframing the question ◮ Rashevsky (1960’s) [9] showed using a network story Minimal volume calculation Blood networks that power output of heart should scale as M 2 / 3 River networks ◮ West et al. (1997 on) [16, 2] managed to find M 3 / 4 Distributed Sources Facility location (a mess—super long story—see previous course...) Size-density law Cartograms ◮ Banavar et al. [1] attempt to derive a general result for References all natural branching networks ◮ Again, something of a mess [2] ◮ We’ll look at and build on Banavar et al.’s work... Frame 16/85
Supply Networks Simple supply networks Introduction Optimal branching Murray meets Tokunaga Single Source History Reframing the question ◮ Banavar et al., Minimal volume calculation Blood networks Nature, River networks (1999) [1] Distributed Sources ◮ Very general Facility location Size-density law Cartograms attempt to find References most efficient transportation networks. Frame 17/85
Supply Networks Simple supply networks Introduction Optimal branching Murray meets Tokunaga ◮ Banavar et al. find ‘most efficient’ networks with Single Source History Reframing the question P ∝ M d / ( d + 1 ) Minimal volume calculation Blood networks River networks ◮ ... but also find Distributed Sources Facility location V blood ∝ M ( d + 1 ) / d Size-density law Cartograms References ◮ Consider a 3 g shrew with V blood = 0 . 1 V body ◮ ⇒ 3000 kg elephant with V blood = 10 V body ◮ Such a pachyderm would be rather miserable. Frame 18/85
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