本文研究了一个具有$N$结点的随机图，其中结点$j$的度为$D_j$，而$\{D_j\}_{j=1}^N$是独立同分布的.使用$\prob(D_j\leq x)=F(x)$。我们假设对于某些$τ>3$和某些常数$c>0$，$1-F(x)\leq c x^{-τ+1}$。该图模型是所谓的配置模型的变体，并且包括具有有限方差的重尾度。当$N\to \infty$时，研究了两个任意连接节点之间的最小边数，也称为图距离或跳数。我们证明了当对数的底等于$ν=\expec[D_j(D_j -1)]/\expec[D_j]>1$时，图的距离像$\log_νN$一样增长。这证实了Newman、Strogatz和Watts \cite{NSW00}的启发式论证。此外，随机波动的渐近平均$\log_ν{N}$的特点，并一致有界。特别是，我们表明收敛分布的中心图的距离沿指数增长的连续性。
英文摘要：
In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with $\prob(D_j\leq x)=F(x)$. We assume that $1-F(x)\leq c x^{-τ+1}$ for some $τ>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance.
The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_νN$, when the base of the logarithm equals $ν=\expec[D_j(D_j -1)]/\expec[D_j]>1$. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_ν{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.
We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviors are possible. When the weights are a.s. larger than a constant, the weight and number of edges in the graph grow proportionally to loglog(n), as for the graph distances. On the other hand, when the continuous-time branching process describing the first passage…
We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent tau in (2,3). In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices. We investigate the case when the speeds are equal and typical distances in a follow-up paper.
We obtain new bounds for the solution of the variance-gamma (VG) Stein equation that are of the correct form for approximations in terms of the Wasserstein and Kolmorogorov metrics. These bounds hold for all parameters values of the four parameter VG class. As an application we obtain explicit Wasserstein and Kolmogorov distance error bounds in a six moment theorem for VG approximation of double Wiener-Itô integrals.
We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ of parameter $p$. When $p>p_c$, there exists almost surely a unique infinite cluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, we prove that the variance of the graph distance in $\mathcal C_p$ between two points of $\mathcal C_p$ is sublinear. The main result extends the works of Benjamini, Kalai and Schramm, Benaim and Rossignol and Damron, Hanson and Sosoe for the study of the variance of passage times in first passage percolation without moment conditions on the edge-weight distribution.
For arbitrary two probability measures on real d-space with given means and variances (covariance matrices), we provide lower bounds for their total variation distance. In the one-dimensional case, a tight bound is given.
In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the variances of Kf*(COCn) and Kf+(COCn) for a random cyclooctane chain based on the known results of others' paper and they are asymptotically subject to normal distribution.
We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We prove that the weight of the optimal path has tight fluctuations around the asymptotical mean of the graph-distance if and only if the following condition holds: the random variable X is such that the continuous-time branching process describing first-passage percolation exploration in the same graph with excess edge weight X has a positive probability to reach …
We consider the problem of estimating species trees from unrooted gene tree topologies in the presence of incomplete lineage sorting, a common phenomenon that creates gene tree heterogeneity in multilocus datasets. One popular class of reconstruction methods in this setting is based on internode distances, i.e. the average graph distance between pairs of species across gene trees. While statistical consistency in the limit of large numbers of loci has been established in some cases, little is known about the sample complexity of such methods. Here we make progress on this question by deriving …
In this paper, we analyze the behavior of the global clustering coefficient in scale free graphs. We are especially interested in the case of degree distribution with an infinite variance, since such degree distribution is usually observed in real-world networks of diverse nature. There are two common definitions of the clustering coefficient of a graph: global clustering and average local clustering. It is widely believed that in real networks both clustering coefficients tend to some positive constant as the networks grow. There are several models for which the average local clustering coeff…
We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent $τ\in (2,3)$. In this model two colors spread with a fixed and equal speed on the unweighted random graph. We analyse how many vertices the two colors paint eventually. We show that coexistence sensitively depends on the initial local neighborhoods of the source vertices: if these neighborhoods are `dissimilar enough', then there is no coexistence, and the `loser' color paints a polynomial fraction of the vertices with a rando…
We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate $λ_1$ ($λ_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $λ_1=λ_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variabl…
In this paper, we present a detailed analysis of the global clustering coefficient in scale-free graphs. Many observed real-world networks of diverse nature have a power-law degree distribution. Moreover, the observed degree distribution usually has an infinite variance. Therefore, we are especially interested in such degree distributions. In addition, we analyze the clustering coefficient for both weighted and unweighted graphs. There are two well-known definitions of the clustering coefficient of a graph: the global and the average local clustering coefficients. There are several models prop…
The average nearest neighbor degree (ANND) of a node of degree $k$ is widely used to measure dependencies between degrees of neighbor nodes in a network. We formally analyze ANND in undirected random graphs when the graph size tends to infinity. The limiting behavior of ANND depends on the variance of the degree distribution. When the variance is finite, the ANND has a deterministic limit. When the variance is infinite, the ANND scales with the size of the graph, and we prove a corresponding central limit theorem in the configuration model (CM, a network with random connections). As ANND prove…
Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multi-type branching process. We find closed-form analytic expressions for the mean…
We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree $0$ (terminal nodes) and outdegree $1$ in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree $0$ and $1$ asymptotically follow a two-dimensional Gaussian law via multivariate martingale methods. We also study t…
We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable.
We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centred, of a random matrix with a variance profile and the standard Gaussian random variable. The second order Poincaré inequality type result is used to establish the bound. Using this bound we prove Central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate variance prof…
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree $\leq n$ in subintervals $\left [ a,b\right ] $ of the support of the underlying orthogonality measure $μ$. We show that, as $n\rightarrow \infty $, this variance is asymptotic to $cn$, for some explicit constant $c>0$.
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the e…
Under natural conditions, we proved the exponential ergodicity in Wasserstein distance of two-type continuous-state branching processes in Lévy random environments with immigration. Furthermore, we expressed accurately the parameters of the exponent. The coupling method and the conditioned branching property play an important role in the approach. Using the tool of superprocesses, the ergodicity in total variance distance is also proved.
In this paper, we prove a local limit theorem for the ratio of the Poisson distribution to the Gaussian distribution with the same mean and variance, using only elementary methods (Taylor expansions and Stirling's formula). We then apply the result to derive an upper bound on the Le Cam distance between Poisson and Gaussian experiments, which gives a complete proof of the sketch provided in the unpublished set of lecture notes by Pollard (2010), who uses a different approach. We also use the local limit theorem to derive the asymptotics of the variance for Bernstein c.d.f. and density estimato…
The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states. We show that all eigenvectors are extended and that the spectral density exhibits a logarithmic o…
