Physical realizability of small-world networks

Supplementing a lattice with long-range connections effectively models small-world networks characterized by a high local and global interconnectedness observed in systems ranging from society to the brain. If the links have a wiring cost associated to their length l, the corresponding distribution q(l) plays a crucial role. Uniform length distributions have received most attention despite indications that q(l) ~ l^{-\alpha} exist, e.g. for integrated circuits, the Internet and cortical networks. While length distributions of this type were previously examined in the context of navigability, we here discuss for such systems the emergence and physical realizability of small-world topology. Our simple argument allows to understand under which condition and at what expense a small world results.