We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes in this model when considered at criticality. In particular, we obtain stronger upper bounds (with respect to those available in the literature) for the probability of observing unusually large maximal clusters, and simplify the arguments needed to derive polynomial upper bounds for the probability of observing unusually small largest components.
We consider inhomogeneous Erdős-Rényi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its centred version. We prove that these extreme eigenvalues are governed at first order by the largest degrees and, for the adjacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which in particular rules out their convergence to a nondegenerate point process. Together with the compa…
Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse random intersection graphs we establish a first order asymptotic for the order of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is an average of nonextinction probabilities of a related multi-type Poisson branching process.
We analyze the component evolution in inhomogeneous random intersection graphs when the average degree is close to 1. As the average degree increases, the size of the largest component in the random intersection graph goes through a phase transition. We give bounds on the size of the largest components before and after this transition. We also prove that the largest component after the transition is unique. These results are similar to the phase transition in Erdős-Rényi random graphs; one notable difference is that the jump in the size of the largest component varies in size depending on the …
Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained (see previous work by Bhamidi, van der Hofstad and van Leeuwaarden). It was proved that when the degrees obey a power law with exponent in the interval (3,4), the sequence of clusters ordered in decreasing size and scaled appropriately converges as n goes to infinity to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that th…
Given a graph $G$ and a constant $γ\in [0,1]$, let $ω^{(γ)}(G)$ be the largest integer $r$ such that there exists an $r$-vertex subgraph of $G$ containing at least $γ\binom{r}{2}$ edges. It was recently shown that $ω^{(γ)}(G)$ is highly concentrated when $G$ is an Erdős-Rényi random graph (Balister, Bollobás, Sahasrabudhe, Veremyev, 2019). This paper provides a simple method to extend that result to a setting of inhomogeneous random graphs, showing that $ω^{(γ)}(G)$ remains concentrated on a small range of values even if $G$ is an inhomogeneous random graph. Furthermore, we give an explicit ex…
Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C x^(2/alpha)) for x > 1 and large n with alpha = 5/48 and C > 0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d > 2.
We identify the scaling limits for the sizes of the largest components at criticality for inhomogeneous random graphs when the degree exponent $τ$ satisfies $τ>4$. We see that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme to study the critical behavior in inhomogeneous random graphs of so-called rank-1 initiated in \cite{Hofs09a}.
We continue the study of the maximum of the scale-inhomogeneous discrete Gaussian free field in dimension two. In this paper, we consider the regime of weak correlations and prove the convergence in law of the centred maximum to a randomly shifted Gumbel distribution. In particular, we obtain limiting expressions for the random shift. As in the case of variable speed branching Brownian motion, the shift is of the form CY, where C is a constant that depends only on the variance at the shortest scales, and Y is a random variable that depends only on the variance at the largest scales. Moreover, …
We analyse the scaling limit of the sizes of the largest components of the Random Intersection Graph $G(n,m,p)$ close to the critical point $p=\frac{1}{\sqrt{nm}}$, when the numbers $n$ of individuals and $m$ of communities have different orders of magnitude. We find out that if $m \gg n$, then the scaling limit is identical to the one of the \ER Random Graph (ERRG), while if $n \gg m$ the critical exponent is similar to that of Inhomogeneous Random Graphs with heavy-tailed degree distributions, yet the rescaled component sizes have the same limit in distribution as in the ERRG. This suggests …
We consider the random hyperbolic graph model introduced by [KPK + 10] and then formalized by [GPP12]. We show that, in the subcritical case $α$ > 1, the size of the largest component is n^{1/(2$α$)+o(1)} , thus strengthening a result of [BFM15] which gave only an upper bound of n^{1/$α$+o(1)}.
We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent τ. We investigate the case where $τ\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(τ-2)/(τ-1)}$, converge to hitting times of a "thinned" Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted …
Consider the inhomogeneous Erdős-Rényi random graph (ERRG) on $n$ vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$, where $(r_n)_{n\in\mathbb{N}}$ is a sequence of graphons converging to a reference graphon $r$. As a generalization of the celebrated LDP for ERRGs by Chatterjee and Varadhan (2010), Dhara and Sen (2019) proved a large deviation principle (LDP) for a sequence of such graphs under the assumption that $r$ is bounded away from 0 and 1, and with a rate function in the form of a low…
This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1\times\cdots\times n_d$, it is shown that the expectation of its largest singular value is upper bounded by $\sqrt {n_1}+\cdots+\sqrt {n_d}$. For the expectation of the largest $\ell^d$-singular value, it is upper bounded by $2^{\frac{d-1}{2}}\prod_{j=1}^{d}n_j^{\frac{d-2}{2d}}\sum^d_{j=1}n_j^{\frac{1}{2}}$. We also derive the upper bounds of the expectations of the largest Z-/H-($\ell^d$)…
In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side result, one shows by virtue of numerical…
Random K-out graphs are receiving attention as a model to construct sparse yet well-connected topologies in distributed systems including sensor networks, federated learning, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes differ in resources and requirements, inhomogeneous random K-out graphs, denoted by $H(n;μ,K_n)$, were proposed recently. Motivated by practical settings where establishing links is costly and only a bounded choice of $K_n$ is feasible ($K_n = O(1)$), we study the size of the largest connected sub-network of …
Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices
Harbin Institute of Technology（哈尔滨工业大学）；Nanyang Technological University（南洋理工大学）；Jinan University（暨南大学）；National University of Singapore（新加坡国立大学）；Changchun University of Science and Technology（长春理工大学）
Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E} X^3<\infty$. Given types of all vertices, an edge exists between vertices $i$ and $j$ independent of anything else and with probability $\min \{1, \frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical phase, which is known to take place when ${\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the asymptotic joint distributions of component sizes of the …
We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. The edge probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W, in which case the proportion of vertices with degree k is close to the probability that a Poisson random var…
In this paper we study the diameter of Inhomogeneous random graphs $G(n,κ,p)$ that are induced by irreducible kernels $κ$. The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erdős-Rényi model $G(n,p)$ for several ranges of $p$. We find upper and lower bounds for the diameter of $G(n,κ,p)$ in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.
The upper tail problem for the largest eigenvalue of the Erdős--Rényi random graph $\mathcal{G}_{n,p}$ is to estimate the probability that the largest eigenvalue of the adjacency matrix of $\mathcal{G}_{n,p}$ exceeds its typical value by a factor of $1+δ$. In this note we show that for $δ>0$ fixed, and $p \rightarrow 0$ such that $n^{\frac{1}{2}} p \rightarrow \infty$, the upper tail probability for the largest eigenvalue of $\mathcal{G}_{n,p}$ is $$\exp\left[-(1+o(1)) \min\left\{\tfrac{(1+δ)^2}{2}, δ(1+δ) \right\} n^{2}p^{2}\log (1/p)\right].$$ In the same regime of $p$, we show that the seco…
We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result appl…
In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes whic…
We apply here methods of inhomogeneous random graphs to a class of random distance graphs. This provides an example outside of the rank 1 models which is still solvable as long as the largest connected component is concerned. In particular, we show that some random distance graphs behave exactly as the classical Erdős-Rényi model not only in the supercritical phase (as was already known) but in the subcritical case as well.
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well a…
The size of the largest common subtree (maximum agreement subtree) of two independent uniform random binary trees on $n$ leaves is known to be between orders $n^{1/8}$ and $n^{1/2}$. By a construction based on recursive splitting and analyzable by standard "stochastic fragmentation" methods, we improve the lower bound to order $n^β$ for $β= \frac{\sqrt{3} - 1}{2} = 0.366$. Improving the upper bound remains a challenging problem.
On the Largest Singular Values of Random Matrices with Independent Cauchy Entries
利用行列式方法研究了具有独立柯西元的大型$ m \times n $实矩形随机矩阵的最大奇异值的分布。我们表明，统计性质（重新缩放的$ \frac{1}{m^2\*n^2}$的一个因素）最大的奇异值同意在极限与强度$ \frac{1}π x^{-3/2} $的非齐次泊松随机点过程的统计，因此，是不同的Tracy-Widom定律。在我们的方法的其他推论，我们显示了一个有趣的数学期望之间的联系复杂的矩形$ m \times n $标准Wishart合奏和真正的矩形$ 2m \times 2n $标准Wishart合奏的决定因素。
For the Erdős-Rényi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.
Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+δ$. Going beyond the Erdős-Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochast…
We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve [1] Arnaudon Thalmaier, C., in order to generalize Harnack inequality with power -for inhomogeneous heat equation - introduced by F.Y Wang. In the case of Ricci ow, we will derive on-diagonal bound of the Heat kernel along Ricci ow (and also for the usual Heat kernel on complete M…
