Decomposing Degree Assortativity in Sparse Spatial Networks

Abstract

Spatial networks are typically assortative: well-connected nodes link to other well-connected nodes, and the usual reading is sorting, popular nodes seeking each other out. In space there is a rival explanation: nearby nodes draw on the same pool of potential neighbors, so their degrees move together even when popularity and location are unrelated. This paper asks when the two can be told apart from network data, and gives a three-part answer. First, in a sparse random-connection model in which latent popularity and position are coupled by a copula, we prove that the limiting assortativity splits exactly into an intensity-covariance channel and a shared-neighbor channel; because the first mixes sorting with density effects, we define the sorting contribution as the increment produced by the dependence when mean degree and transitivity are held fixed, exactly zero at independence. Second, we prove that three observable summaries (mean degree, transitivity, assortativity) identify the model globally, including on the boundary of no dependence, by a computer-assisted proof in exact rational arithmetic that also verifies the conditions for valid inference; simulations confirm the asymptotics. Third, on data the machinery acts as a gate: no sorting estimate is reported unless the model first fits. In eleven metropolitan location-based networks the model class is rejected and direct estimates find essentially no dependence; the observed assortativity is evidence of neither sorting nor shared opportunity. In a national co-authorship network the verdict reverses: productive authors concentrate where researcher density is high, yet graph-only attribution is again refused, the misfit pointing to the team structure of multi-author papers.

Version 2