  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 a multiplex network with a
fast layer embedded in a slow one. To connect a pair of points, one can then
use either the fast or slow layer, or both, with a switching cost when going
from one layer to the other. We consider here distributions of points in spaces
of arbitrary dimension d and search for the fast-layer network of given size
that minimizes the average time to reach a central node. We discuss the d = 1
case analytically and the d > 1 case numerically, and show the existence of
transitions when we vary the network size, the switching cost and/or the
relative speed of the two layers. Surprisingly, there is a transition
characterized by a 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. Our findings underscore the importance of
considering switching costs while studying optimal network structures, as small
variations of the cost can lead to strikingly dissimilar results. Finally, we
discuss real-world subways and their efficiency for the cities of Atlanta,
Boston, and Toronto. We find that real subways are farther away from the
optimal shapes as traffic congestion increases.
