Hub Neighbor-Degree Diagnostics for Sparse Random Graphs
arXiv:2607.26624
The paper introduces a diagnostic based on the average degree of neighbors of hub vertices to assess the fit of sparse random graph models, providing theoretical limits and goodness‑of‑fit tests for various network models.
Abstract
Networks with nearly identical degree distributions can place their hubs in sharply different neighborhoods. We develop a model diagnostic based on the mean degree of the neighbors of a degree- vertex. Under rank-one inhomogeneous random graphs, this statistic has degree-invariant centering and fluctuations. Under non-rank-one kernels, posterior uncertainty about the root type can instead determine both centering and scale. Under linear preferential attachment, the statistic grows as . We turn these model-specific limits into goodness-of-fit tests for specified sparse-graph nulls and a weighted log-degree slope test for residual hub-neighborhood trends. Simulations evaluate null calibration, degree-distribution misspecification, and power against degree-matched preferential-attachment alternatives. Applications to high-school contact and arXiv coauthorship networks show that the method separates level misspecification from disassortative and positive residual trends. Reddit interaction networks provide a further appendix example.