In many physical networks, including neurons in the brain1,2, three-dimensional integrated circuits3 and underground hyphal networks4, the nodes and links are physical objects that cannot intersect or overlap with each other. To take this into account, non-crossing conditions can be imposed to constrain the geometry of networks, which consequently affects how they form, evolve and function. However, these constraints are not included in the theoretical frameworks that are currently used to characterize real networks5–7. Most tools for laying out networks are variants of the force-directed layout algorithm8,9—which assumes dimensionless nodes and links—and are therefore unable to reveal the geometry of densely packed physical networks. Here we develop a modelling framework that accounts for the physical sizes of nodes and links, allowing us to explore how non-crossing conditions affect the geometry of a network. For small link thicknesses, we observe a weakly interacting regime in which link crossings are avoided via local link rearrangements, without altering the overall geometry of the layout compared to the force-directed layout. Once the link thickness exceeds a threshold, a strongly interacting regime emerges in which multiple geometric quantities, such as the total link length and the link curvature, scale with the link thickness. We show that the crossover between the two regimes is driven by the non-crossing condition, which allows us to derive the transition point analytically and show that networks with large numbers of nodes will ultimately exist in the strongly interacting regime. We also find that networks in the weakly interacting regime display a solid-like response to stress, whereas in the strongly interacting regime they behave in a gel-like fashion. Networks in the weakly interacting regime are amenable to 3D printing and so can be used to visualize network geometry, and the strongly interacting regime provides insights into the scaling of the sizes of densely packed mammalian brains.A modelling framework is presented to determine the optimal layout and physical properties of networks in which the nodes and links have physical sizes and intersections between components is prohibited.

中文翻译：

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物理网络的结构转型

在许多物理网络中，包括大脑中的神经元 1、2、三维集成电路 3 和地下菌丝网络 4，节点和链接是物理对象，彼此不能交叉或重叠。考虑到这一点，可以施加非交叉条件来约束网络的几何形状，从而影响它们的形成、演变和功能。然而，这些约束不包括在当前用于表征真实网络的理论框架中 5-7。大多数用于布置网络的工具都是力导向布局算法 8,9 的变体——它假定节点和链接是无量纲的——因此无法揭示密集物理网络的几何形状。在这里，我们开发了一个建模框架，该框架考虑了节点和链接的物理大小，允许我们探索非交叉条件如何影响网络的几何形状。对于小链接厚度，我们观察到一种弱交互机制，其中通过局部链接重新排列避免了链接交叉，与力导向布局相比，不会改变布局的整体几何形状。一旦链接厚度超过阈值，就会出现一个强相互作用机制，其中多个几何量，例如链接总长度和链接曲率，随链接厚度缩放。我们表明，两个机制之间的交叉是由非交叉条件驱动的，这使我们能够分析地推导出过渡点，并表明具有大量节点的网络最终将存在于强交互机制中。我们还发现，弱相互作用机制中的网络对压力表现出类似固体的反应，而在强相互作用机制中，它们表现得像凝胶一样。弱交互机制中的网络适用于 3D 打印，因此可用于可视化网络几何形状，强交互机制提供了对密集哺乳动物大脑大小缩放的见解。提出了一个建模框架来确定最佳布局禁止节点和链路具有物理尺寸和组件之间有交叉的网络的物理特性。