On the structural properties of small-world networks with finite range of shortcut links

TL;DR

This paper introduces a variant of Small-World Networks with a parameter controlling shortcut range, revealing that small-world properties can emerge without global shortcuts and analyzing how average path length behaves.

Contribution

It proposes a new short-range SWN model with a tunable shortcut length scale and derives an expression for average path length through simulations and analytical methods.

Findings

Small-world phenomenon occurs for small r, even without global shortcuts.

Average path length in short-range SWNs is nonmonotonic with system size.

Derived an analytical approximation for average path length in short-range SWNs.

Abstract

We explore a new variant of Small-World Networks (SWNs), in which an additional parameter ($r$) sets the length scale over which shortcuts are uniformly distributed. When $r=0$ we have an ordered network, whereas $r=1$ corresponds to the original SWN model. These short-range SWNs have a similar degree distribution and scaling properties as the original SWN model. We observe the small-world phenomenon for $r \ll 1$ indicating that global shortcuts are not necessary for the small-world effect. For short-range SWNs, the average path length changes nonmonotonically with system size, whereas for the original SWN model it increases monotonically. We propose an expression for the average path length for short-range SWNs based on numerical simulations and analytical approximations.