Developmental time windows for spatial growth generate multiple-cluster small-world networks

Many networks extent in space, may it be metric (e.g. geographic) or non-metric (ordinal). Spatial network growth, which depends on the distance between nodes, can generate a wide range of topologies from small-world to linear scale-free networks. However, networks often lacked multiple clusters or communities. Multiple clusters can be generated, however, if there are time windows during development. Time windows ensure that regions of the network develop connections at different points in time. This novel approach could generate small-world but not scale-free networks. The resulting topology depended critically on the overlap of time windows as well as on the position of pioneer nodes.

💡 Research Summary

The paper introduces a novel network‑growth framework that couples spatial distance‑dependent attachment with a temporal “time‑window” mechanism, thereby generating spatial networks that simultaneously exhibit multiple community clusters and small‑world properties.
Model definition. Nodes are embedded in three‑dimensional Euclidean space. The probability of forming an edge between a newly added node U and an existing node V is the product of three factors: (i) a distance‑dependent term (P_{dist}=βe^{-γd(U,V)}) (with β=6, γ=6), (ii) the temporal activity of U’s time window (P_{w(U)}^{time}(t)), and (iii) the temporal activity of V’s time window (P_{w(V)}^{time}(t)). The time‑window functions are bell‑shaped, centered at a node‑class specific mean (μ_i=i/(k+1)) (where k is the number of windows) and scaled so that the integral over the interval