Spatial Dependence in Directed Preferential-Attachment Networks

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.