Central Limit Type Theorems in the Generalized Random Graphs with Random Vertex Weights (original) (raw)

Central limit theorems and statistical inference for some random graph models

2015

Random graphs and networks are of great importance in any fields including mathematics, computer science, statistics, biology and sociology. This research aims to develop statistical theory and methods of statistical inference for random graphs in novel directions. A major strand of the research is the development of conditional goodness-of-fit tests for random graph models and for random block graph models. On the theoretical side, this entails proving a new conditional central limit theorem for a certain graph statistics, which are closely related to the number of two-stars and the number of triangles, and where the conditioning is on the number of edges in the graph. A second strand of the research is to develop composite likelihood methods for estimation of the parameters in exponential random graph models. Composite likelihood methods based on edge data have previously been widely used. A novel contribution of the thesis is the development of composite likelihood methods based ...

Central Limit Theorem for Fluctuations of Linear Eigenvalue Statistics of Large Random Graphs. Dilut

We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with "non-gaussian" part of the Wigner ensemble variance. C 2012 American Institute of Physics. [http://dx.

Quenched Central Limit Theorems for the Ising Model on Random Graphs

Journal of Statistical Physics, 2015

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by √ N for the Ising model on random graphs with N vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime β > β c or β > 0 and B = 0, where β is the inverse temperature, β c is the critical inverse temperature and B is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in Z d . For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs: Diluted regime

Journal of Mathematical Physics, 2012

We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with "non gaussian" part of the Wigner ensemble variance.

Approximating the Statistics of various Properties in Randomly Weighted Graphs

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

Consider the setting of randomly weighted graphs, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the nonnegative reals. Under this setting, properties of weighted graphs typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some properties albeit the problem of computing them in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the diameter of a given weighted graph, yet, computing the expected diameter of a given randomly weighted graph is #P-hard even if the edge weights are identically distributed.

Computing the Expected Values of some Properties of Randomly Weighted Graphs

Consider the setting of randomly weighted graphs, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the nonnegative reals. Under this setting, properties of weighted graphs typically become random variables and we are interested in computing their statistical features. Unfortunately, this turns out to be computationally hard for some properties albeit the problem of computing them in the traditional setting of algorithmic graph theory is tractable. For example, there are well known efficient algorithms that compute the diameter of a given weighted graph, yet, computing the expected diameter of a given randomly weighted graph is #P-hard even if the edge weights are identically distributed.

Annealed limit theorems for the Ising model on random regular graphs

The Annals of Applied Probability, 2019

In a recent paper, Giardinà et al. [ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121-161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.

Poisson convergence of numbers of vertices of a given degree in random graphs

Discussiones Mathematicae Graph Theory, 1996

The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.

Asymptotic Normality of Some Graph-Related Statistics

Journal of Applied Probability, 1989

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.