O. Khorunzhiy
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.
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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.
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.
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.
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 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 consider the ensemble of real symmetric random matrices $H^{(n,ρ)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $ρ/n$. We prove that the normalized eigenvalue counting function of $H^{(n,ρ)}$ weakly converges in average as $n,ρ\to\infty$ and $ρ=o(n^α)$ for any $α>0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.
In this paper we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_t:t\in [0,1])$, where initially we start with a critical Erdős-Rényi graph ER(n, 1/n), and then evolve forwards in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval $[0, 1]$ is of order $n^{2/3} log^{1/3} n$ with high probability. This is in contrast to the largest component in the static critical Erdős-Rényi graph, which is of order $n^{2/3}$.
In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, can one develop efficient numerical schemes to estimate the probability of an Erdős-Rényi graph containing an excessively large number of a fixed given subgraph? Using the large deviation principle we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring…
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。
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.
In this paper we define a directed percolation over Erdos-Renyi graphs and derive weak limit results for the resulting stochastic process.
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。
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.
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.
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…
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。
Web crawlers are used by internet search engines to gather information about the web graph. In this paper we investigate a simple process which models such software by walking around the vertices of a graph. Once initial random vertex weights have been assigned, the robot crawler traverses the graph deterministically following a greedy algorithm, always visiting the neighbour of least weight and then updating this weight to be the highest overall. We consider the maximum, minimum and average number of steps taken by the crawler to visit every vertex of firstly, complete k-partite graphs and se…
We prove expectation and concentration results for the following random variables on an Erdős-Rényi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probabilit…
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 $p^3N^2\to +\infty$.
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。
We consider the bulk eigenvalue statistics of Laplacian matrices of large Erdős-Rényi random graphs in the regime $p \geq N^δ/N$ for any fixed $δ>0$. We prove a local law down to the optimal scale $η\gtrsim N^{-1}$ which implies that the eigenvectors are delocalized. We consider the local eigenvalue statistics and prove that both the gap statistics and averaged correlation functions coincide with the GOE in the bulk.
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 $p^3N \to \infty$.
Concentration of Hitting Times in Erdös-Rényi graphs
本文证明Erdős-Rényi图中随机游走击中时间高度集中，可由边数、度与距离的简单公式近似，误差极小。
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^…
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 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 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 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…
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 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 $c_3 < p^c_{\text{eff}}n\log n < c_2.$
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。
Homomorphic-core phase transition threshold in Erdős--Rényi random graphs
该研究确定了Erdős–Rényi随机图成为同态核的相变阈值，推导了子图同构问题的紧ETH下界。
A conditional compound Poisson process approach to the sparse Erdős-Rényi random graphs: moderate deviations
提出条件复合Poisson过程表示稀疏Erdős-Rényi随机图的连通分量，揭示其相变与零程模型凝聚转变的联系，并在非临界状态下推导了中等偏差原理。
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 $G(n,M)$。
For two independent Erdős-Rényi graphs $\mathbf G(n,p)$, we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product, we prove that the maximal overlap is asymptotically $\frac{n}{2α-1}$ for $p=n^{-α}$ with some constant $α\in (1/2,1)$.
Fluctuations of functions of sparse Erdős-Rényi graphs
本文研究稀疏 Erdős-Rényi 图重标度邻接矩阵函数对角元的谱涨落，证明其渐近分布为两个独立高斯变量之和，并揭示了在谱尺度 p 处发生的相变。
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…
Limiting empirical spectral distribution for the non-backtracking matrix of an Erdős-Rényi random graph
本文精确描述了Erdős-Rényi图非回溯矩阵在$np/\log n\to\infty$时的极限经验谱分布，通过部分去随机化简化谱，并借助Tao-Vu替换原理和Bauer-Fike定理证明其与原谱接近。
The critical Karp--Sipser core of Erdős--Rényi random graphs
针对Erdős--Rényi随机图$\mathrm{G}[n, λ/n]$的Karp--Sipser核，证明临界状态（$λ=\mathrm{e}$）下核大小为$n^{3/5}$阶，证实猜想并给出渐近规律与分布描述，方法基于微分方程法。
The Erdős-Rényi process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph models have been introduced and studied. In this paper we study the so-called Bohman-Frieze process, a simple modification of the Erdős-Rényi process. The Bohman-Frieze process begins with an empty graph on…
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: …
Let $N_{\triangle}(G)$ be the number of triangles in a graph $G$. In [14] and [25] (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs $G_m\sim G(n,m)$: \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ\left(δ^2n^3\right)\right) \qquad \text{if $n^{-3/2}\ll δ\ll n^{-1}$} \] and \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ(δ^{2/3}n^2) \right) \qquad \text{if $n^{-3/4} \ll δ\ll 1$.} \] Neeman, Radin an…
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…
Color-avoiding percolation on the Erdős-Rényi random graph
本文研究常数平均度的随机染色 Erdős-Rényi 随机图中的避色渗流，刻画最大 CA 分量呈线性、对数及有界大小的三种状态。
We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erdös-Rényi random graph. By analyzing finite graphs of increasing volume, as well as the graphon variational problem in the infinite volume limit, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.
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…
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。
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…
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…
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 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.
On Concentration Inequality of the Laplacian Matrix of Erdős-Rényi Graphs
本文研究Erdős-Rényi随机图归一化拉普拉斯矩阵谱范数的集中性质，得到了正则化拉普拉斯矩阵的最优界，并利用度数的一致集中性质建立了齐次情形下的一致集中不等式。
Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$
本文研究稀疏 Erdős-Rényi 随机图中三角形计数的中偏差，给出两种模型下上尾对数概率的常数因子估计，并在非局部化区域获得 $G(n,p)$ 的渐近精确结果。
Meeting and coalescence times for random walks in the largest component of the Erdős-Rényi random graph
研究厄多斯-雷尼随机图最大连通分量上随机游走的相遇和合并时间，通过证明相遇时间界限并结合比较不等式，得出预期合并时间和完全选民模型共识在严格超临界、轻微超临界和临界区域都为\(n\)阶。
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…
Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph
本文研究了 Erdős-Rényi 图在特定平均度区间内拉普拉斯算子的谱隙，证明其最小非零特征值以高概率由仅通过一条边连接巨连通分量的规模为 $t$ 的线状小子图决定，并给出了精确的渐近表达式。
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