Limiting distributions of triangle counts in linear preferential attachment models

We derive distributional approximations for the number of triangles in the linear preferential attachment model $\mathrm{P}\mathrm{A}\mathrm{M}(m,\delta )$, where $m\ge 2$ and $\delta >-m$, with explicit rates of convergence. The limiting distribution undergoes a phase transition from Gaussian to another nontrivial distribution, which we characterize explicitly. The asymptotic behavior is governed by the interplay between the hidden random environment and the mean-field interaction effect. In particular, our analysis also yields a continuous phase transition in the expected number of triangles as $\delta$ varies.