We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $Ω$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,λ)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λV^{1/3}$, where $λ$ is fixed and positive. We show that for any such model, there is a phase transition at $p_c$ analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold $p_c$. In particular, we show that the largest cluster inside a scaling window of size $|p-p_c|=Θ(\cn^{-1}V^{-1/3})$ is of size $Θ(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(ε^{-2}\log(Vε^3))$, with $ε=\cn(p_c-p)$. We also obtain an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order $Θ(\cn(p-p_c))$. Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the $n$-cube and certain Hamming cubes, as well as the spread-out $n$-dimensional torus for $n>6$.
In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The lat…
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.
We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In Borgs et al., Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann. Probab. (2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2…
Let $H$ be a fixed graph and $\mathcal{G}$ a subcritical graph class. In this paper we show that the number of occurrences of $H$ (as a subgraph) in a uniformly at random graph of size $n$ in $\mathcal{G}$ follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations. As a case study, we get explicit expressions for the number of triangles and cycles of length four for the family of series-parallel graphs.
Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when $p$ is at least about $\sqrt{(3/2)\ln n/n}$, and not below this unless $p$ is very small.) We give two related proofs of this statement, together with a relatively simple proof of a fundamental "stability" theorem for triangle-free subgraphs of $G_{n,p}$, originally due to Kohayakawa, Łuczak and Rödl, tha…
We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λ2^{n/3}$, where $λ$ is a small positive constant. Let $ε=n(p-p_c(n))$. In two previous papers, we showed that the largest cluster inside a scaling window given by $|ε|=Θ(2^{-n/3})$ is of size $Θ(2^{2n/3})$, below this scaling window it is at most $2(\log2) nε^{-2}$, and above this scaling window it is at most $O(ε2^n)$. In this paper, we prove that for $p - p_c(n) \geq e^…
We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question of Babai, Simonovits and Spencer (Journal of Graph Theory, 1990). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with $M$ edges, where $M >> n$, is ``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ v…
The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.
We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $ np- \log n \to - \infty$ the probability that the graph is connected goes to $0$, while when $ np- \log n \to + \infty$ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $ np- \log n \to t \in \mathbb{R}$. We find that th…
We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraphs counts in the Erd{\H o}s-Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering and improving recent results derived for triangles as well as results using the Wasserstein distance.
Sampling uniform simple graphs with power-law degree distributions with degree exponent $τ\in(2,3)$ is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triang…
In his work \cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called \emph{Stein-Tikhomirov method}. \citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. \citet*{R…
