Optimal Transport Flows for Distributed Production Networks

Network flows often exhibit a hierarchical treelike structure that can be attributed to the minimization of dissipation. The common feature of such systems is a single source and multiple sinks (or vice versa). In contrast, here we study networks with only a single source and sink. These systems can arise from secondary purposes of the networks, such as blood sugar regulation through insulin production. Minimization of dissipation in these systems leads to vascular shunting, a single vessel connecting the inlet and outlet. We show instead how optimizing the transport time yields network topologies that match those observed in the insulin-producing pancreatic islets. These are patterns of periphery-to-center and center-to-periphery flows. The obtained flow networks are broadly independent of how the flow velocity depends on the flow flux, but continuous and discontinuous phase transitions appear at extreme flux dependencies. Lastly, we show how constraints on flows can lead to buckling of the branches of the network, a feature that is also observed in pancreatic islets.

Figure 1

Vasculature of pancreatic islets. Idealized center-to-periphery flow (a), left-right symmetric flow (b), and periphery-to-center flow (c). (d) Microscopy of vasculature, from Ref. [10], showing the high degree of tortuousness in the vasculature of pancreatic islets. (e) The network structure considered. This is constructed by taking a circular subsection of a hexagonal grid of nodes. Neighbors are induced from a Delaunay triangulation. Each edge has a conductivity $C_e$ associated to it, which are the parameters to be optimized over. The inlet and outlet are shown by arrows.

Figure 2

Optimized network structures. (a) Minimal dissipation network with sinks at all nodes. (b) Minimal dissipation network with a single sink at the edge. (c) Per-node time-minimizing network with a single sink at the edge. Cell colors indicate the average time to the outlet(s). Flow lines are colored by pressure, their thickness indicating the (square root of) flow magnitude.

Figure 3

Phase transition in triangle geometry. (a) Triangle geometry. We consider $S_{\text{source}}=1=F_1+F_2$. (b) Energy ($⟨T⟩$) landscape for various $\delta$. Circles indicate select minima. (c) Value of $F_1$ obtained by minimizing $⟨T⟩$ for various scalings $\delta$. There is a discontinuous transition at $\delta \approx 0.275$ and a continuous transition at $\delta =1$.

Figure 4

Varying the importance $\alpha$ between time from nodes to the outlet and the inlet to nodes. (a) The left-right symmetric solution obtained at $\alpha =0.5$. (b) The solution in Fig. 2 is reversed for $\alpha =1.0$. Background color indicates $T^{1-\alpha }{T}_r^{\alpha }$.

Figure 5

Optimal graphs under minimal flow constraints. (a) At ${\mathcal{F}}_m=0.042$, the network in Fig. 2 loses its front collection channels. (b) The collection point starts moving to the right as ${\mathcal{F}}_m$ is increased to 0.062. (c) At ${\mathcal{F}}_m=0.073$, the collection point has moved to the far right. (d) One horizontal vessel has been removed by introducing kinks at $\mathcal{F}=0.078$, which increases the flow in the remaining vessels. (e) Buckling increases; ${\mathcal{F}}_m=0.125$. As ${\mathcal{F}}_m\to 1$, the global optimum becomes a Hamiltonian path through the nodes. (f) In dark blue (left axis), the average time $⟨T⟩$ is shown as a function of ${\mathcal{F}}_m$. The average flow through the nodes $⟨\mathcal{F}⟩$ is shown in light brown (right axis). The dashed line shows $f(x)=x$, which matches the slope of $⟨\mathcal{F}⟩$ in the buckling section. Background colors denote section: global optimum in blue, unbuckled solution in green, and buckled in red.