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