Sep 2026· Advances in Applied Probability· pp. 1-36· 0 citations· 17 references
TL;DR
For clique counts, a multivariate quantitative normal approximation result through the Malliavin–Stein method is established and distributional convergence based on a central limit theorem for sequences of associated random variables is obtained.
Abstract
We study normal approximation of subgraph counts in a model of spatial scale-free random networks known as the
age-dependent random connection model
. In the light-tailed regime where only moments of order
left parenthesis 2 plus epsilon right parenthesis
(
2
+
ε
)
$(2+\varepsilon)$
are finite, we study the asymptotic normality of both clique and subtree counts. For clique counts, we establish a multivariate quantitative normal approximation result through the Malliavin–Stein method. In the more delicate case of subtree counts, we obtain distributional convergence based on a central limit theorem for sequences of associated random variables.
Homogeneous random graphs, also known as Erd{\H os}-R\'enyi graphs, are a subset of the family of dense random graphs, specified by a graphon. We analyze several goodness-of-fit tests for these models that are based on subgraph counts. To obtain the limiting null distribution of the test statistics we use a decompositi...
We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional s...
Johannes Bäumler, Lukas Lüchtrath, C. Mönch· 0 citations
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We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps...
Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is clos...
This work introduces BERGM Elastic Net, an adaptive empirical-Bayes approach that combines lasso shrinkage with ridge stabilization in a Bayesian ERGM that is developed for over-specified network models containing many related structural and covariate effects.
Dan Han, Vicki Modisette, Tinghan Li et al.· 0 citations
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