我们扩展的拼图渗流模型来分析图形，其中底层的人和拼图图是Erdens-Rényi随机图。令$p_{\text{ppl}}$和$p_{\text{puz}}$表示边存在于相应人物和谜题图中的概率，并定义$p_{\text{eff}}= p_{\text{ppl}}p_{\text{puz}}$，即有效概率。我们证明了对于常数$c_1>1$和$c_2> π^2/6$和$c_3<e^{-5}$，如果$min(p_{\text{ppl}},p_{\text{puz}}) > c_1 \log n /n$的临界有效概率$p^c_{\text{eff}}$，满足$c_3 < p^c_{\text{eff}}n\log n < c_2.$
英文摘要：
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 $c_3 < p^c_{\text{eff}}n\log n < c_2.$
Upper exponential inequalities for the tail probabilities of the centered and normalized number of triangles in the Erdös-Rényi graph are obtained, where the probability of every edge is fixed. The result is formulated in terms of random fields.
We analyze Ising/Curie-Weiss models on the (directed) Erdős-Rényi random graph on $N$ vertices in which every edge is present with probability $p$. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime $β<1$ when $p=p(N)$ satisfies $p^3N^2\to +\infty$.
We study the exploration of an Erdös-Rényi random graph by a respondent-driven sampling method, where discovered vertices reveal their neighbours. Some of them receive coupons to reveal in their turn their own neighbourhood. This leads to the study of a Markov chain on the random graph that we study. For sparse Erdös-Rényi graphs of large sizes, this process correctly renormalized converges to the solution of a deterministic curve, solution of a system of ODEs absorbed on the abscissa axis. The associated fluctuation process is also studied, providing a functional central limit theorem, with a…
We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph $G(N,p)$. We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime $β>1$ and the case when an external magnetic field is present. In both cases, we assume that $p=p(N)$ satisfies $p^3N \to \infty$.
We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with $N$ vertices and edge probability $p=p(N)$ that were introduced by Bovier and Gayrard [J.\ Statist.\ Phys., 72(3-4):643--664, 1993] and investigated in two previous articles by the authors. We prove Central Limit Theorems for the partition function of the model and -- at other decay regimes of $p(N)$ -- for the logarithmic partition function. We find critical regimes for $p(N)$ at which the behavior of the fluctuations of the partition function changes.
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 present results on the concentration properties of the spectral norm $\|A_p\|$ of the adjacency matrix $A_p$ of an Erdős-Rényi random graph $G(n,p)$. First we consider the Erdős-Rényi random graph process and prove that $\|A_p\|$ is uniformly concentrated over the range $p\in [C\log n/n,1]$. The analysis is based on delocalization arguments, uniform laws of large numbers, together with the entropy method to prove concentration inequalities. As an application of our techniques we prove sharp sub-Gaussian moment inequalities for $\|A_p\|$ for all $p\in [c\log^3n/n,1]$ that improve the general…
As suggested by Itai Benjamini, we introduced a variant of the Erdös- Rényi random graph process with a forbidden degree $k$, in which every edge adjacent to a vertex $v$ is removed when the degree of $v$ reaches $k$ (but the removed edges may very well be added again later). We study the existence of a giant component, depending on the forbidden degree $k$ and the time parameter $t$. We prove that for $k\textgreater{}4$ a giant component appears at some point, while for $k\textless{}4$, a giant component never occurs. The main tool of our study is the local limit of the random graph process: …
We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdős-Rényi graph $G(N,p)$. Tracy-Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in [17,46]. We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the…
We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation …
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…
In [18] we analysed a simple undirected random graph subject to constraints on the total number of edges and the total number of triangles. We considered the dense regime in which the number of edges per vertex is proportional to the number of vertices. We showed that, as soon as the constraints are \emph{frustrated}, i.e., do not lie on the Erdős-Rényi line, there is breaking of ensemble equivalence, in the sense that the specific relative entropy per edge of the \emph{microcanonical ensemble} with respect to the \emph{canonical ensemble} is strictly positive in the limit as the number of ver…
We investigate Random Sequential Adsorption (RSA) on a random graph via the following greedy algorithm: Order the $n$ vertices at random, and sequentially declare each vertex either active or frozen, depending on some local rule in terms of the state of the neighboring vertices. The classical RSA rule declares a vertex active if none of its neighbors is, in which case the set of active nodes forms an independent set of the graph. We generalize this nearest-neighbor blocking rule in three ways and apply it to the Erdős-Rényi random graph. We consider these generalizations in the large-graph lim…
We introduce a variant of the Erdös--Rényi random graph where the number of vertices is random and follows a Poisson law. A very simple Markov property of the model entails that the Lukasiewicz exploration is made of \textit{independent} Poisson increments. Using a vanilla Poisson counting process, this enables us to give very short proofs of classical results such as the phase transition for the giant component or the connectedness for the standard Erdös--Rényi model.
A probabilistic proof of cutoff in the Metropolis algorithm for the Erdős-Rényi random graph
我们研究了Metropolis算法对超立方体上的分布的混合，该分布对应于边缘概率为p的Erdens-Rényi随机图。该马尔可夫链在窗口大小为n的max{p，1-p} n log n处截止，Diaconis和Ram（2000）使用傅里叶分析证明了这一结果。在这里，我们给出了一个替代的证明，依赖于耦合和投影到一个二维马尔可夫链。这样做是希望概率技术将更容易推广到不太对称分布。我们还描述了这个模型的大都市和吉布斯采样之间的密切关系。我们的证明扩展到边缘概率随n变化的情况。在这种情况下，我们还表明，一个自然的坐标明智的耦合是尖锐的，当且仅当边缘概率的顺序为1/n。
We consider a dynamic Erdős-Rényi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $λ$ and switches off at rate $μ$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that startin…
We model the transmission of information of a message on the Erdös-Rény random graph with parameters $(n,p)$ and limited resources. The vertices of the graph represent servers that may broadcast a message at random. Each server has a random emission capital that decreases by one at each emission. We examine two natural dynamics: in the first dynamics, an informed server performs its attempts, then checks at each of them if the corresponding edge is open or not; in the second dynamics the informed server knows a priori who are its neighbors, and it performs all its attempts on its actual neighb…
Let $\mathcal{G}(N,\frac 1Nt_N)$ be the Erdős-Rényi graph with connection probability $\frac 1Nt_N\sim t/N$ as $N\to\infty$ for a fixed $t\in(0,\infty)$. We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of $\mathcal{G}(N,\frac 1Nt_N)$, registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order $N$), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreov…
In this paper we are concerned with the SIR model with random vertex weights on Erdős-Rényi graph $G(n,p)$. The Erdős-Rényi graph $G(n,p)$ is generated from the complete graph $C_n$ with $n$ vertices through independently deleting each edge with probability $(1-p)$. We assign i. i. d. copies of a positive r. v. $ρ$ on each vertex as the vertex weights. For the SIR model, each vertex is in one of the three states `susceptible', `infective' and `removed'. An infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices. An infec…
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.
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…
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).
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^…
We consider a number $ν_n$ of components in a random graph $G(n,p)$ with $n$ vertices, where the probability of an edge is equal to $p$. By operating with special generating functions we shows the next asymptotic relation for factorial moments of $ν_n$: $$ \mathsf{E}(ν_n-1)^{\underline s} = (1+o(1))\left( \frac 1p \sum\limits_{k=1}^\infty\frac{k^{k-2}}{k!}(npq^n)^k\right)^s + o(1) $$ as $n$ tends to $\infty$ and $q=1-p$. And the following inequations hold: $$ 1-2nq^{n-1} \le p_n\le\frac{1}{nq^n}, $$ $$ 1-\frac{1}{nq^n}\le pi_n\le nq^{n-1}, $$ where $p_n$ is the probability that $G(n,p)$ is con…
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…
We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $μ\ll 1/n$, and is then declared open with probability $p = λ/n$ where $λ> 1$. We study a random walk on this dynamical environment which jumps at rate $1/n$ along every open edge. We show that the mixing time of the full system exhibits cutoff at $\log n/μ$. We do this by showing that the random walk component mixes faster than the environment process; along the way, we control the time it takes for the walk to become isolated.
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: $$μ_{2} (A(G))\leq 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_{1}+n_{2})p}).$$
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…
The largest components of the critical Erdős-Rényi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size $an^{2/3}$ for large $a$. Our results, which extend work of Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$ around $1/n$. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.
The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph $G(n,m)$. Our approach is based on applying Freedman's inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related $G(n,p)$ model.
Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the t…
Starting with the large deviation principle (LDP) for the Erdős-Rényi binomial random graph $\mathcal{G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph $\mathcal{G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in $\mathcal{G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random…
Almost Sure Central Limit Theorems and the Erdos-Renyi law for Expanding Maps of the Interval
对区间上的一大类扩张映射，证明了Lipschitz观测量的部分和满足几乎必然的中心极限定理（ASCLT）.事实上，我们在Kantorovich度量中提供了收敛速度。部分和的最大值也服从ASCLT。关键工具是最近得到的一个指数不等式。然后，我们得到了Lipschitz观测量的移动平均的上确界的几乎必然收敛速度（Erdos-Renyi型定律）。最后，我们应用熵估计ASCLT的，refi ne Shannon-McMillan-Breiman和Ornstein-Weiss定理。
