How loops form in transport networks: from electrical discharges to jellyfish canal network

What is it about?

Nature offers us a wide spectrum of spatial, transport networks, from our own blood vessels to electrical discharges in the sky during a storm. Such networks take various shapes. They can have a tree-like geometry, where branches of the network only split and repel each other as they grow. In other cases, where branches attract and reconnect, we deal with looping structures. Networks with many loops are widespread in living organisms, where they actively transport oxygen or nutrients and remove metabolic waste products. An important advantage of looping networks is their reduced vulnerability to damage - in networks without loops, the destruction of one branch can cut off all connected branches, whereas in networks with loops, there is always another connection to the rest of the system. Many transport networks grow in response to a diffusive field, such as the concentration of a substance or the pressure in the system. In such systems, only competition between branches and repulsion had previously been observed. However, we found that when a branch of the network reaches the system boundary, the interactions between branches change drastically. Previously repelling branches begin to attract each other, leading to the sudden formation of loops. The same phenomenon has been observed in a number of systems: in fracture dissolution experiments, mimicking cave formation; in the Saffman-Taylor experiment where a less viscous fluid is injected into a more viscous one and the boundary between the two fluids breaks into finger-like patterns; in electrical discharges; and even in the growth of a gastrovascular canal network of the jellyfish Aurelia. In our work, we present a model describing the interactions between branches in systems driven by diffusive fluxes. We focus on how these interactions change when a branch approaches the system boundary and breaks through. Competition and repulsion between branches then disappear, and attraction appears. This inevitably leads to loop formation. Our model predicts that the attraction between neighboring branches after breakthrough occurs regardless of the network geometry or the details of the transport processes in the system.

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Photo by Garth Manthe on Unsplash

Photo by Garth Manthe on Unsplash

Why is it important?

We have shown that loops can form near breakthrough even in systems where branches are strongly competing, previously thought impossible. This explains why this phenomenon is so widespread in physical and biological systems. Our explanation of the common underlying mechanism behind the phenomenon significantly advances the understanding of how looping transport networks dynamically emerge.

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