Spatial preferential attachment with choice-based edge step
Yury Malyshkin
TL;DR
This work analyzes a spatial preferential attachment model augmented with a two-stage, choice-based edge step, incorporating two auxiliary samples to model long-range connections. The authors derive evolution laws for the top in-degrees and establish a three-regime behavior governed by the quantity $a+d\\alpha$: subcritical growth $M_1(n)\\sim n^{a+d\\alpha}$, critical scaling $M_1(n)\\ln n / n\\to 2/(d\\alpha)^2$, and supercritical convergence $M_k(n)/n\\to x_k^{\\ast}$ for $k\\le K$, where $K$ is the largest positive root of a fixed-point system. The convergence in the supercritical regime is proved via stochastic approximation, using an induction on $k$ and bounding arguments around the fixed point. These results illuminate condensation and degree concentration phenomena arising from geometry and the power of choice in network growth.
Abstract
We study the asymptotic behavior of the maximal in-degree in the spatial preferential attachment model with a choice-based edge step. We prove different types of behavior of maximal in-degree based on the model's parameters.