  We show analytically that abrupt structural transitions can arise in
functionally optimal networks, driven by small changes in the level of
transport congestion. Our findings are based on an exactly solvable model
system which mimics a variety of biological and social networks. Our results
offer an explanation as to why such diverse sets of network structures arise in
Nature (e.g. fungi) under essentially the same environmental conditions. As a
by-product of this work, we introduce a novel renormalization scheme involving
`cost motifs' which describes analytically the average shortest path across
multiple-ring-and-hub networks.
