Investigation of triangle counts in graphs evolving by clustering attachment

The clustering attachment (CA) model proposed by Bagrow and Brockmann in 2013 may be used as an evolution tool for undirected random networks. A general definition of the CA model is introduced. Theoretical results are obtained for a new CA model that can be treated as the former’s limit in the case of the model parameters $\alpha\to0$ and $\epsilon = 0$. This study is focused on the triangle count of connected nodes at an evolution step $n$, an important characteristic of the network clustering considered in the literature. As is proved for the new model below, the total triangle count $\Delta n$ tends to infinity almost surely as $n\to\infty$ and the growth rate of $E\Delta_n$ at an evolution step $n\geqslant2$ is higher than the logarithmic one. Computer simulation is used to model sequences of triangle counts. The simulation is based on the generalized Pólya–Eggenberger urn model, a novel approach introduced here for the first time.