Saharon Shelah
Affirming a conjecture of Erdős and Renyi we prove that for any (real number) c_1>0 for some c_2>0, if a graph G has no c_1(log n) nodes on which the graph is complete or edgeless (i.e. G exemplifies |G| not-> (c_1 log n)^2_2) then G has at least 2^{c_2n} non-isomorphic (induced) subgraphs.
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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 consider a natural variant of the Erdős-Rényi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by the combinatorial data fusion problem that itself is connected to a number of important problems in graph theory. We will show that a phase transition occurs when the number of special vertices is roughly $n^{1/3}$, where $n$ is the number of vertices.
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: $$μ_{2} (A(G))\leq 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_{1}+n_{2})p}).$$
Given a graph $Γ= (V, E)$ on $n$ vertices and $m$ edges, we define the Erdős-Rényi graph process with host $Γ$ as follows. A permutation $e_1,\dots,e_m$ of $E$ is chosen uniformly at random, and for $t\leq m$ we let $Γ_t = (V, \{e_1,\dots,e_t\})$. Suppose the minimum degree of $Γ$ is $δ(Γ) \geq (1/2 + \varepsilon)n$ for some constant $\varepsilon > 0$. Then with high probability, $Γ_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two. Given $0\leq p\leq 1$ we let $Γ_p$ be the Erdős-Rényi subgraph of $Γ$, obtained by retaining each edge independently with prob…
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.
For a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $χ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $χ(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.
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 $n^{2/3}$.
The process of fluctuations of the giant component of an Erdős-Rényi graph
本文研究 Erdős-Rényi 图过程中巨连通分量在平均度从 1 增至无穷时的演化，并证明其重标度涨落收敛于一个具有显式协方差的高斯过程。
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。
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…
We consider the near-critical Erdős-Rényi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+λ/n^{4/3}$ and $A$ is large, \[\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{λA^2}{2}-\frac{λ^2A}{2}}\] where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows $A$ and $λ$ to depend on $n$. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula spec…
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.
From Erdos-Renyi graphs to Linial-Meshulam complexes via the multineighbor construction
研究Erdos-Renyi随机图的$m$-邻居复形，发现其以高概率呈现特定维数和面分布，并猜想该测度与Linial-Meshulam测度在总变差下收敛。
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…
In the shotgun assembly problem for a graph, we are given the empirical profile for rooted neighborhoods of depth $r$ (up to isomorphism) for some $r\geq 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erdős-Rényi $\mathcal G(n, \fracλ{n})$, we show that the shotgun assembly threshold $r_* \approx \frac{ \log n}{\log (λ^2 γ_λ)^{-1}}$ where $γ_λ$ is the probability for two independent Poisson-Galton-Watson trees with parameter $λ$ to be rooted isomorphic with each other. Our result sharpens a constant factor in a previous work by Mossel and Ross…
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…
Clique structure and other network properties of the tensor product of Erdős-Rényi graphs
该研究分析Erdős-Rényi图张量积的团数量与最大团大小，研究扩展聚类系数及与局部效率的关系，分析孤立顶点数，并探讨随机图模积与最大公共子图问题的关联。
In this work we perform computational and analytical studies of the Randić index $R(G)$ in Erdös-Rényi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $ξ\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $ξ< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < ξ< 10$ (where $0<R(G)< n/2$), and a regime of almost complete graphs for $ξ> 10$ ($…
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 $G(n,M)$。
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。
Erd\H os and Rényi claimed and Vu proved that for all $h \ge 2$ and for all $ε> 0$, there exists $g = g_h(ε)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$ elements of $A$ is bounded by $g$, and such that $|A \cap [1,x]| \gg x^{1/h - ε}$. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a $g$ that satisfies $g_h(ε) \ll ε^{-1}$, improving the bound $g_h(ε) \ll ε^{-h+1}$ obtained by Vu. Finally we use the "alte…
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。
Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of containing $\lfloor k/2 \rfloor$ Hamilton cycles, and one perfect matching if $k$ is odd, all edge-disjoint. With an eigenvalue condition on $P$, and conditions on its row sums, $G_{n, P}\in \mathcal{A}_k$ happens with high probability if and only if $G_{n, P}$ has minimum degree $k$ whp. We also provide a hitting time version. As a special case, the random …
Sandwiching between random regular graphs and Erdős-Rényi graphs: configuration model and unions of perfect matchings
本文通过引入中间模型链的耦合框架，验证了 Kim-Vu 夹逼猜想在大度数情形并证明弱化版本，建立了随机正则图与 Erdős-Rényi 图之间的新联系。
The intersection of a random geometric graph with an Erdős-Rényi graph
本文研究随机几何图与 Erdős-Rényi 图的交集图，在边概率和半径均趋于零时分析其团数、独立数、连通性、哈密顿性、色数和直径等性质。
Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of $C_4$-free graphs are the Erdős-Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Rényi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Rényi orthogonal polarity graph.
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…
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…
For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0\leq q^*(G) < 1$. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erdős-Rényi random graph $G_{n,p}$ with $n$ vertices and edge-probability $p$. Two key findings are that the modularity is $1+o(1)$ with…
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…
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。
Graph matching is an important problem in machine learning and pattern recognition. Herein, we present theoretical and practical results on the consistency of graph matching for estimating a latent alignment function between the vertex sets of two graphs, as well as subsequent algorithmic implications when the latent alignment is partially observed. In the correlated Erdős-Rényi graph setting, we prove that graph matching provides a strongly consistent estimate of the latent alignment in the presence of even modest correlation. We then investigate a tractable, restricted-focus version of graph…
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。
For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish the leading order of the exponential rate function for the probability that the number of copies of $H$ in the Erdős--Rényi graph $G(n,p)$ exceeds its expectation by a factor $1+u$, assuming $n^{-κ(H)}\ll p\ll1$, with $κ(H) = 1/(2Δ)$, where $Δ\ge 1$ is the maximum degree of $H$. This improves on a previous result of Chatterjee and the second author, who obtained $κ(H)=c/(Δ|E|)$ for a constant $c>0$. Moreover, for the case of cycle counts we can take $κ$ as large as $1/2$. We additionally obtain the sharp upper tail for Scha…
Fluctuations of the Ising free energy on Erdős-Rényi graphs
本文研究了在埃尔多斯-雷尼随机图上伊辛模型的自由能波动，确定了不同条件下配分函数的极限分布特性。
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