A. Khorunzhy
We generalise the Erdos-Renyi limit theorem on the maximum of the partial sums of random variables to the case when the number of terms in these sums is randomly distributed. Certain relations between the limiting theorems of this type and the spectral theory of random graphs and random matrices are discussed.
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Using recent results on the concentration of the largest eigenvalue and maximal vertex degree of large random graphs, we show that the infinite sequence of Erd\H os-Rényi random graphs $G(n,ρ_n/n)$ such that $ρ_n/\log n$ infinitely increases as $n\to\infty$ verifies a version of the graph theory Riemann Hypothesis.
In this paper we define a directed percolation over Erdos-Renyi graphs and derive weak limit results for the resulting stochastic process.
Limit theorems for Randic index for Erdős-Renyi graphs
该研究针对Erdős-Renyi模型的图，证明了广义Randic指数在稀疏与稠密两种情形下，当顶点数趋于无穷时均集中于其均值附近，完善了相关图论指数的极限理论。
Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdos-Renyi random graph of a given degree.
We study the asymptotic behaviour of Markov processes on large weighted Erdos-Renyi graphs where the transition rates of the vertices are only influenced by the state of their neighbours and the corresponding weight on the edges. We find the ratio of vertices being in a certain state will converge to the solution of a differential equation obtained from mean field approximation if the graph is dense enough, namely, the average degree is at least of order $N^{\frac{1}{2}+ε}$. Proof for convergence in probability in the transient regime is shown.
We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erdős-Rényi random graphs. Let $p_{\text{ppl}}$ and $p_{\text{puz}}$ denote the probability that an edge exists in the respective people and puzzle graphs and define $p_{\text{eff}}= p_{\text{ppl}}p_{\text{puz}}$, the effective probability. We show for constants $c_1>1$ and $c_2> π^2/6$ and $c_3<e^{-5}$ if $min(p_{\text{ppl}},p_{\text{puz}}) > c_1 \log n /n$ the critical effective probability $p^c_{\text{eff}}$, satisfies $c_3 < p^c_{\text{eff}}n\log n < c_2.$
Let $A \in \mathbb{R}^{n \times n}$ be the adjacency matrix of an Erdős Rényi graph $G(n, d/n)$ for $d = ω(1)$ and $d \leq 3\log(n)$. We show that as $n$ goes to infinity, with probability that goes to $1$, the adjacency matrix of the $3$-core of $G(n, d/n)$ is invertible.
Functional Central Limit Theorem for the principal eigenvalue of dynamic Erdős-Rényi random graphs
本文研究动态 Erdős-Rényi 随机图在顶点数增大时邻接矩阵主特征值的演化，证明其满足泛函中心极限定理，并揭示主特征值继承单边动力学特征。
We obtain a functional Erd\H os-R\' enyi law of large numbers for "nonconventional" sums of the form $\Sig_n=\sum_{m=1}^nF(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of exponentially fast $ψ$-mixing random vectors and $F$ is a Borel vector function extendin in several directions our previous result concerning i.i.d. random variables $X_1,X_2,...$.
We bound the error for the normal approximation of the number of triangles in the Erdos-Renyi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein-bounds obtained by Barbour, Karonski and Rucinski (1989), resolving a long-standing open problem. The proofs are based on a new variant of the Stein-Tikhomirov method - a combination of Stein's method and characteristic functions introduced by Tikhomirov (1980).
In this work we prove that the giant component of the Erdös--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $δ$-hyperbolic for any positive $δ$ with probability tending to one as $n\to\infty$. As a corollary we provide an alternative proof that the giant component of $G(n,c/n)$ when c>1 has zero spectral gap almost surely as $n\to\infty$.
We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent critical state.
We analyse the eigenvalues of Erdös--Rényi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=Ω(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=Ω(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim G(n_{1},n_{2},p)$. We show that with probability tending to $1$ as $n_{1}$ tends to infinity: $$μ_{2} (A(G))\leq 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_{1}+n_{2})p}).$$
We address the issue of the Central Limit Theorem for (both local and global) empirical measures of diffusions interacting on a possibly diluted Erdős-Rényi graph. Special attention is given to the influence of initial condition (not necessarily i.i.d.) on the nature of the limiting fluctuations. We prove in particular that the fluctuations remain the same as in the mean-field framework when the initial condition is chosen independently from the graph. We give an example of non-universal fluctuations for carefully chosen initial data that depends on the graph. A crucial tool for the proof is t…
On torsion in eulerian magnitude homology of Erdos-Renyi random graphs
本文通过引入欧拉Asao–Izumihara复形并建立其剥脱消失阈值，确定了Erdős–Rényi随机图欧拉数量同调群无挠的参数区间。
Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs
本文证明稀疏Erdos-Renyi图上临界Ising模型的混合时间以高概率为多项式阶，结合随机局部化框架与树上Glauber动力学经典结果处理异常局部邻域。
The $N$ vertices of a quantum random graph are each a circle independently punctured at Poisson points of arrivals, with parallel connections derived through for each pair of these punctured circles by yet another independent Poisson process. Considering these graphs at their critical parameters, we show that the joint law of the re-scaled by $N^{2/3}$ and ordered sizes of their connected components, converges to that of the ordered lengths of excursions above zero for a reflected Brownian motion with drift. Thereby, this work forms the first example of an inhomogeneous random graph, beyond th…
On the Asymptotics of the Connectivity Probability of Erdos-Renyi Graphs
本文通过非齐次随机游走分析，将Erdos-Renyi图连通概率问题转化为泊松增量随机游走在桥条件下的游程概率，从而刻画其精确渐近行为。
For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in proving central limit theorems. This paper introduces a method for establishing central limit theorems in the sparse graph setting. It works for problems which display a key property which has been variously called "endogeny", "long-range independence", and "replica symmetry" in the literature. Examples of such problems are maximum weight matching, $λ$-diluted…
What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.
For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdos-Renyi type law). We end up with an application to entropy estimation ASCLT's that refi ne Shannon-McMillan-Breiman and Ornstein-Weiss theorem…
Functional Central Limit Theorem for the simultaneous subgraph count of dynamic Erdős-Rényi random graphs
本文研究了动态Erdős-Rényi随机图中子图计数向量的函数中心极限定理，通过适当的中心化和缩放，证明其收敛于多维高斯过程。
Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions in the cases of constant a…
Fluid limit and gelation in the frozen Erdős-Rényi random graph
本文研究了冻结的Erdős-Rényi随机图，通过冻结含唯一环的连通分量演化以阻止巨分量产生，利用Wormald微分方程方法分析了主要统计量的流体极限，并精确描述了图完全冻结时间的渐近行为。
We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-m…
We study the antiferromagnetic Potts model on the Poissonian Erdös-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.
We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^…
Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of \ER random graphs $G(n, p_n)$, where $p_n = n^{-α}$ for $0 < α< 1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-$1$ neighborhoods, $G$ is exactly reconstructable for $0 < α< \frac{1}{3}$, but not reconstructable for $\frac{1}{2} < α< 1$. Given the collection of distance-$2$ neighborhoods, $G$ is ex…
Consider the complete graph on \(n\) vertices where each edge is independently open with probability \(p,\) or closed otherwise. Phase transitions for such graphs for \(p = \frac{C}{n}\) have previously been studied using techniques like branching processes and random walks. In this paper, we use an alternate component counting argument for establishing phase transition and obtaining estimates on the sum size of the non-giant components. As a corollary, we also obtain estimates on the size of the giant component for \(C\) large: If \(C\) is sufficiently large, there is a positive constant \(M_…
A Method for Generating Connected Erdos-Renyi Random Graphs
本文提出将图探索过程与非齐次 Poisson 随机游走耦合的精确采样算法，可在线性时间内生成稀疏连通 $G(n,p)$，并扩展至 $G(n,M)$。
We address the following foundational question: what is the population, and sample, Frechet mean (or median) graph of an ensemble of inhomogeneous Erdos-Renyi random graphs? We prove that if we use the Hamming distance to compute distances between graphs, then the Frechet mean (or median) graph of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix of the ensemble. We show that the result also holds for the sample mean (or median) when the population expected adjacency matrix is replaced with the sample mean adjacency matrix. Consequently, the F…
For $x\in [0,1],$ the run-length function $r_n(x)$ is defined as the length of the longest run of $1$'s amongst the first $n$ dyadic digits in the dyadic expansion of $x.$ Erdős and Rényi proved that $\lim\limits_{n\to\infty}\frac{r_n(x)}{\log_2n}=1$ for Lebesgue almost all $x\in[0,1]$. Let $H$ denote the set of monotonically increasing functions $φ:\mathbb{N}\to (0,+\infty)$ with $\lim\limits_{n\to\infty}φ(n)=+\infty$. For any $φ\in H$, we prove that the set \[ E_{\max}^φ=\left\{x\in [0,1]:\liminf\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=0, \limsup\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=+\infty…
We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph. This graph is constructed on $N$ vertices and every edge has probability $p$ to be present. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993] and analyzed by the authors in a previous note, in which we consider the case of $p=p(N)$ satisfying $p^3N^2\to +\infty$ and $β<1$. In the current note we prove a quenched Central Limit Theorem for the magnetization for $p$ satisfying $pN \to \infty$ in the high-temperature regime $β<1$. We also show a non-standard Central Limit Theorem for $p^4N^…
Consider a critical Erdös-Rényi random graph: $n$ is the number of vertices, each one of the $\binom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{-1}+λn^{-4/3}$, $λ$ being a fixed real number. When $n$ goes to infinity, Addario-Berry, Broutin and Goldschmidt have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distribution to a continuous limit $\mathcal{G}_λ$ made of random real graphs. In this paper, we consider notably the dynamical percolation on critical Erdös-Rén…
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of models, the nature of this emergence is similar to that of the Erdős-Rényi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals. To date, (a) has been proven for a numbe…
Reaction networks are commonly used within the mathematical biology and mathematical chemistry communities to model the dynamics of interacting species. These models differ from the typical graphs found in random graph theory since their vertices are constructed from elementary building blocks, i.e., the species. In this paper, we consider these networks in an Erd\H os-Rényi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero, which is a property of great interest in the reaction network community. Specifically, if the number of s…
Shotgun assembly of unlabeled Erdos-Renyi graphs
该研究证明当密度参数$p_n$处于$n^{-1/2}\log^C n$到常数$c$区间时，无标号Erdős–Rényi图可由其顶点1-邻域多重集唯一重构，确立了$\tilde\Theta(n^{-1/2})$为重构性转变区间，解决了Gaudio和Mossel的相关问题。
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