Phase transition in preferential attachment-detachment through embedding
Michael Hinz, Angelica Pachon
TL;DR
This work analyzes a directed random graph with preferential edge attachment and detachment, revealing a phase transition in the tail of the in-degree distribution as network growth and decay compete. By embedding the graph dynamics into a generalized Yule model, the authors obtain a complete description of the limit in-degree distribution across regimes: power-law tails in the supercritical case, exponential tails in the subcritical case, and strictly intermediate decay at criticality, with regime thresholds determined by the rates \lambda_1, \lambda_2, and \mu_2. The embedding yields explicit asymptotics and closed-form representations for the limit probabilities p_j using Beta and hypergeometric functions, and extends classical Yule dynamics to incorporate detachment. The results provide a unified, analytically tractable framework to understand how edge loss interacts with preferential attachment, with potential applications to evolving networks where links can fail or disappear.
Abstract
We study a random graph model with preferential edge attachment and detachment through the embedding into a generalized Yule model. We show that the in-degree distribution of a vertex chosen uniformly at random follows a power law in the supercritical regime but has an exponential decay in the subcritical. We provide the corresponding asymptotics. In the critical regime we observe an intermediate decay. The regimes are clearly defined in terms of parameter ranges.