Spatial Dependence in Directed Preferential-Attachment Networks

2026-07-20

AI Summary

This study investigates the coupling between node activity synchronization and preferential attachment mechanisms in spatially embedded directed networks. The authors propose a directed preferential attachment model driven by a spatial Gaussian process, which generates a log-normal field to capture the spatiotemporal persistence of in-degree and out-degree weights. Under the constraint of no self-loops, they analyze the asymptotic behavior of the degree distribution. Innovatively integrating spatial Gaussian processes with directed preferential attachment, they derive a strictly concave inverse mapping for in-degree weights and construct a decomposition framework that separates global traffic from short-range spatial dependencies. Applying sublinear preferential attachment, MM weight estimation, profile likelihood, and spatial pseudo-likelihood methods to transatlantic air transportation networks, they identify a spatial co-exceedance phenomenon at approximately 150 km scales, confirming both distance-decay-dependent transitivity and distance-independent common-mode effects.

Abstract

Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered network histories, we develop a minorization-maximization (MM) weight estimator and profile likelihood for the PA exponent; temporal pre-whitening and a spatial quasi-likelihood estimate the latent covariance. Simulations verify transmission of distance-decaying dependence and show how random segment volume creates a distance-independent common mode in raw degrees. An analysis of U.S. domestic flights (2015-2019) separates network-wide volume variation from a short-range spatial component. An observed-volume reconstruction reproduces the raw-degree common mode, and the fitted field yields an exploratory co-exceedance transition scale of roughly 150 km. A per-carrier analysis of European air traffic also reveals the same decomposition.

Problem

• spatial dependence
• directed networks
• preferential attachment
• Gaussian process
• degree distribution

Innovation

• spatially embedded networks
• directed preferential attachment
• Gaussian process lognormal fields
• minorization-maximization estimation
• co-exceedance scale