Limiting distributions of triangle counts in linear preferential attachment models

Abstract

We derive distributional approximations for the number of triangles in the linear preferential attachment model $\mathrm{PAM}(m,\hat{I}´)$, where $m\ge 2$ and $\hat{I}´>-m$, 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 $\hat{I}´$ varies.

38 pages, 11 figures