A structural transition in physical networks.

Abstract

In many physical networks, from neurons in the brain [1, 2] to 3D integrated circuits [3] or underground hyphal networks [4], the nodes and links are physical objects unable to cross each other. These non-crossing conditions constrain their layout geometry and affect how these networks form, evolve and function, limitations ignored by the theoretical framework currently used to characterize real networks [5, 6, 7, 8, 9, 10]. Indeed, most current network layout tools are variants of the Force-Directed Layout (FDL) algorithm [11, 12], which assumes dimensionless nodes and links, hence are unable to reveal the geometry of densely packed physical networks. Here, we develop a modeling framework that accounts for the physical reality of nodes and links, allowing us to explore how the non-crossing conditions affect the geometry of the network layout. For small link thicknesses, rL, we observe a weakly interacting regime where link crossings are avoided via local link rearrangements, without altering the overall layout geometry. Once rL exceeds a threshold, a strongly interacting regime emerges, where multiple geometric quantities, from the total link length to the link curvature, scale with rL. We show that the crossover between the two regimes is driven by excluded volume interactions, allowing us to analytically derive the transition point, and show that large networks eventually end up in the strongly interacting regime. We also find that networks in the weakly interacting regime display a solid-like response to stress, whereas they behave in a gel-like fashion in the strongly interacting regime. Finally, we show that the weakly interacting regime offers avenues to 3D print networks, while the strongly interacting regime offers insight on the scaling of densely packed mammalian brains.