We typically think of networks as abstract representations of complex systems in which any physicality of their constituents is disregarded. In a number of real networks, however, nodes and links are spatially embedded physical objects that cannot intersect with each other. If the size of nodes and links is small compared to the available space, such as in power grids, physicalitylikely has little effect on the network; however, if the volume of the network is comparable to the volume of the available space, physicality will affect the structure,evolution, andfunction of the networks. Thus, we wonder: when does physicality start mattering? Today, I’ll talk about a tractable random growth model of physical networks that provides insight into this question. Our model describes linear physical networks, where links are non-overlapping straight cylinders. Growth is achieved by sequentially adding nodes to randomly chosen points within the unit cube. The new node connects to a randomly chosen accessible node from the existing network, taking into account non-crossing conditions. We find that, with increasing link thickness, the onset of physicality occurs in stages. In the first stage, we observe changes in the global properties of the system. Physicality is initially manifested as the shortening of the average link length. Curiously, such weakly physical networks have zero-measure volume despite being non-transparent. On the other hand, in strongly physical networks whose volume occupies a finite fraction of the available space, everything becomes local, and we observe changes in network properties, such as the degree distribution and clustering coefficient.
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