Physical realizability of small-world networks

TL;DR

This paper explores the conditions under which small-world networks can physically form, emphasizing the importance of link length distributions and their costs, with implications for systems like the brain and the Internet.

Contribution

It provides a simple framework to understand when and how small-world topology can emerge considering wiring costs and length distributions.

Findings

Small-world networks can form under specific length distribution conditions.

Wiring costs influence the physical realizability of small-world topologies.

Power-law length distributions are relevant for real-world networks like the brain and the Internet.

Abstract

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.