Asymptotic normality for triangle counting in the sparse $β$-model

We study the number of triangles $T_n$ in the sparse $β$-model on $n$ vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin-Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$, as $n\to\infty$.

💡 Research Summary

This paper investigates the statistical behavior of triangle counts in the sparse β‑model, a random graph model that incorporates vertex‑specific parameters to capture degree heterogeneity observed in many real‑world networks. The authors first derive precise asymptotic expressions for the mean and variance of the number of triangles (T_n) in terms of the ℓ‑norms of the heterogeneity vector (\mu=(\mu_1,\dots,\mu_n)^\top) where (\mu_i=e^{\beta_i}>0). Under the sparsity assumptions (\mu_{\max}\to0) and (|\mu|_2\to\infty), they show
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