Weighted dependency graphs

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables. To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vall\'ee on the asymptotic normality of subword counts in random texts generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces

Central limit theorems for patterns in multiset permutations and set partitions

We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses journal layou

Modeling heterogeneity in random graphs through latent space models: a selective review

We present a selective review on probabilistic modeling of heterogeneity in random graphs. We focus on latent space models and more particularly on stochastic block models and their extensions that have undergone major developments in the last five years

Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$\mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf 1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T \subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P(\{i,j\} \in G_N)$ and assume $d_N, N-d_N\underset{N \rightarrow \infty}{\longrightarrow} \infty$. If $\mathfrak C_N(T) =\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$ for any $T$, the standardized empirical eigenvalue distribution of $A_N$ converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additional assumption that $|\frac N 2 - d_N- \eta \sqrt{d_N}| \underset{N \rightarrow \infty}{\longrightarrow} \infty$ for some $\eta>0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.Comment: 21 pages, 7 figure