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

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.

💡 Research Summary

The paper introduces a variant of the classic Watts‑Strogatz small‑world network (SWN) in which the length of shortcut edges is limited to a fraction r of the total number of nodes N. This model, called the Short‑Ranged Small‑World Network (SRSWN), interpolates between a regular one‑dimensional ring lattice (r = 0) and the original SWN (r = 1). The construction proceeds as follows: start with a ring of L = 2N nodes, add shortcuts with probability p (so the expected number of shortcuts is x = Lp). Each shortcut is placed uniformly at random among the r L nearest neighbours of its origin node, i.e., its Euclidean distance never exceeds r N.

Key structural findings

1. Degree distribution – The authors show (in the Appendix) that the degree distribution of SRSWN is identical to that of the original SWN: a narrow, almost Poisson distribution, because the only modification is a distance cutoff, not a change in the number of shortcuts per node.
2. Average shortest‑path length ℓ – Numerical simulations reveal three regimes as the system size L grows: (i) for small L, ℓ grows almost linearly with L, mimicking a regular lattice; (ii) for intermediate L, ℓ reaches a maximum and then decreases with further increase of L, a non‑monotonic behaviour absent in both regular lattices and classic SWNs; (iii) for large L, ℓ resumes a logarithmic increase with L, confirming the emergence of the small‑world effect even when shortcuts are short‑ranged (r ≪ 1).

Analysis of ℓ(n)
ℓ(n) denotes the average shortest‑path distance between two nodes separated by Euclidean distance n on the underlying ring. For SRSWN the authors argue and confirm numerically that:

• When n < r N, shortcuts cannot be used, so ℓ(n) ≈ n (pure lattice behaviour).
• For r N ≤ n ≤ (1 − r) N, ℓ(n) grows linearly with n but with a slope α < 1, reflecting the fact that a walker can typically reach a shortcut after a constant number of steps and then jump within the allowed range.
• Near the ends (n ≈ 0 or n ≈ N) the curve bends due to boundary effects.

Scaling relations
The authors derive a renormalization‑group‑type scaling: ℓ(L λ, p/λ, r) = ℓ(L, p, r)/λ, which implies ℓ = L g(x, r) with x = Lp (the total number of shortcuts). Combining this with a decomposition of the network into a “restricted” part of length R = r L and a “free” part of length L − R leads to the central expression

 ℓ = (L − R) K(y) + R f(y),  y = r Lp,

where f(y) is the known scaling function for the original SWN and K(y) captures the additional linear contribution from the short‑range region. Numerical data show that K(y)≈f(y) − α(y), with α(y) ≈ log(1 + 2y²/24 + y)/(4y).

Limiting behaviours

• Large‑y limit (r L p ≫ 1): ℓ ≈