Spatial symmetry breaking in optimal networks*

Despite its importance for practical applications, not much is known about the optimal shape of a network that connects in an efficient way a set of points. This problem can be formulated in terms of multiplex networks with a fast layer embedded in a slow one. To connect a pair of points, we can then use either the fast or slow layer, or both, with a switching cost when going from one layer to another. We consider the simpler problem of a distribution of points in the plane and ask for the fast layer network of a given length that minimizes the average time to reach a central node. We discuss the 1d case analytically and the 2d case numerically and show the existence of transitions when we vary the network length, the switching cost, and the relative speed of the two layers. Surprisingly, we see a transition characterized by a spatial symmetry breaking indicating that it is sometimes better to avoid serving a whole area in order to save on switching costs, at the expense of using more the slow layer. Moreover, we find that similar transitions are also observed in heterogeneous real-world population densities in cities. Our results underscore the importance of considering switching costs while studying the optimal subway structures, as small variations of the cost can lead to strikingly dissimilar optimal structures. Finally, we discuss real-world subways and their efficiency for Toronto, Boston, and Atlanta. We found that real subways are farther away from the optimal shapes as the car traffic congestion increases.