Investigation of triangle counts in graphs evolved by uniform clustering attachment

TL;DR

A new clustering attachment model which can be considered as the limit of the former clustering attachment model as model parameter $\alpha$ tends to zero is proposed and proved that total triangle count tends to infinity.

Abstract

The clustering attachment model introduced in the paper Bagrow and Brockmann (2013) may be used as an evolution tool of random networks. We propose a new clustering attachment model which can be considered as the limit of the former clustering attachment model as model parameter $α$ tends to zero. We focus on the study of a total triangle count that is considered in the literature as an important characteristic of the network clustering. It is proved that total triangle count tends to infinity a.s. for the proposed model. Our simulation study is used for the modeling of sequences of triangle counts. It is based on the interpretation of the clustering attachment as a generalized Pólya-Eggenberger urn model that is introduced here at first time.

Figures (2)

• Figure 1: The CA random graph $(V_n, E_n)$ with $n=5000$ (left) and $n=10000$ (right): nodes from ${\tilde{V}}_n$ are shown in black, edges from ${\tilde{E}}_n \cap E_n$ are in dark grey and the rest of graph is in light grey.
• Figure 2: Plots $\left\{\left(n, {\bar{\Delta}}_{n}(G_{1, \ell})-\Delta_{1}(G_{1, \ell})\right), \ 0\leqslant n \leqslant 10^6\right\}$ where plots with $\ell\in\{1,2,3\}$ are shown in black, grey and light grey collors, respectively.