scholarly journals On the Lower Tail Variational Problem for Random Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 301-320 ◽  
Author(s):  
YUFEI ZHAO
Keyword(s):  
Variational Problem ◽  
Random Graph ◽  
Random Graphs ◽  
Large Deviation ◽  
Tail Probability ◽  
Edge Density ◽  
Lower Tail ◽  
Lower Tail Probability ◽  
Large Deviation Problem ◽  
Deviation Problem

We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

scholarly journals Cramér type large deviations for trimmed L-statistics

2018 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
Nadezhda Gribkova
Keyword(s):  
Large Deviations ◽  
Weight Function ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Natural Conditions ◽  
New Approach ◽  
Probabilities Of Large Deviations ◽  
Large Deviation Problem ◽  
Deviation Problem

CRAMÉR TYPE LARGE DEVIATIONS FOR TRIMMED L-STATISTICSIn this paper, we propose a new approach to the investigationof asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. 1982 – the first and, as far as we know, the single article where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic with smooth weight function based onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

A Large Deviation Result on the Number of Small Subgraphs of a Random Graph

2001 ◽  
Vol 10 (1) ◽  
pp. 79-94 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Random Graph ◽  
Positive Constant ◽  
Present Author ◽  
Large Deviation ◽  
Tail Probability ◽  
Partial Answer ◽  
Martingale Inequality ◽  
Concentration Result ◽  
Large Deviation Result

Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.[bull ] What is the upper tail probability Pr(YH [ges ] (1 + ε)[ ](YH))?[bull ] For which λ does YH have sub-Gaussian behaviour, namely(formula here)where c is a positive constant?[bull ] Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that(formula here)We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].

Resampling Permutation Probability Values for Six Measures of Qualitative Variation

2007 ◽  
Vol 104 (3) ◽  
pp. 773-776 ◽  
Author(s):  
Janis E. Johnston ◽  
Kenneth J. Berry ◽  
Paul W. Mielke
Keyword(s):  
Tail Probability ◽  
Fortran Program ◽  
Test Statistics ◽  
Test Statistic ◽  
Lower Tail ◽  
Qualitative Variation ◽  
Data Table ◽  
Kendall’S Τ ◽  
Lower Tail Probability

An algorithm and associated FORTRAN program are provided for six common measures of ordinal association: Kendall's τ a and τ b, Stuart's τ c, Goodman and Kruskal's γ, and Somers' dyx and dxy. Program ROMA reports the observed data table, the values for the six test statistics, and the resampling upper- and lower-tail probability values associated with each test statistic.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

scholarly journals CLT-related large deviation bounds based on Stein's method

2007 ◽  
Vol 39 (3) ◽  
pp. 731-752 ◽  
Author(s):  
Martin Raič
Keyword(s):  
Random Graphs ◽  
Finite Population ◽  
Large Deviation ◽  
Random Variables ◽  
Stein’S Method ◽  
Stein's Method ◽  
Population Statistics ◽  
Sums Of Random Variables ◽  
Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

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