  We study network properties of networks evolving in time based on optimal
transport principles. These evolve from a structure covering uniformly a
continuous space towards an optimal design in terms of optimal transport
theory. At convergence, the networks should optimize the way resources are
transported through it. As the network structure shapes in time towards
optimality, its topological properties also change with it. The question is how
do these change as we reach optimality. We study the behavior of various
network properties on a number of network sequences evolving towards optimal
design and find that the transport cost function converges earlier than network
properties and that these monotonically decrease. This suggests a mechanism for
designing optimal networks by compressing dense structures. We find a similar
behavior in networks extracted from real images of the networks designed by the
body shape of a slime mold evolving in time.
