Clustering in a preferential attachment network with triangles

TL;DR

The paper analyzes a two-parameter generalization of affine preferential attachment that injects triangles at random. It develops a rigorous probabilistic framework showing the degree distribution is asymptotically a power law with exponent $\gamma=1+\frac{1}{A}$, where $A=\frac{[(\alpha+1)^2+\delta\alpha]}{(\alpha+1)[2(\alpha+1)+\delta]}$, yielding three regimes corresponding to $\gamma>3$, $\gamma=3$, and $\gamma<3$. It proves the average local clustering coefficient remains positive with high probability, while the global clustering coefficient exhibits regime-dependent decay: $\Theta(1)$ for $A<\tfrac{1}{2}$, $\Theta((\log t)^{-1})$ for $A=\tfrac{1}{2}$, and $\Theta(t^{1-2A})$ for $A>\tfrac{1}{2}$. These results clarify how triangle insertions interact with heavy-tailed degree distributions and highlight the measurement-dependent nature of clustering in evolving networks.

Abstract

We study a generalization of the affine preferential attachment model where triangles are randomly added to the graph. We show that the model exhibits an asymptotically power-law degree distribution with adjustable parameter $γ\in (1,\infty)$, and positive clustering. However, the clustering behaviour depends on how it is measured. With high probability, the average local clustering coefficient remains positive, independently of $γ$, whereas the expectation of the global clustering coefficient does not vanish only when $γ>3$.

Theorems & Definitions (16)

• Theorem 3.1
• Remark 1
• Remark 2
• Theorem 3.2: The average local clustering coefficient does not vanish
• Remark 3
• Remark 4
• Theorem 3.3
• Remark 5
• Lemma 4.1
• proof