Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching $\sqrt{2\over 3} \cosh\sqrt{5\over 18}\approx0.9325$ as $n\to\infty$. The limiting probability that it consists of trees, unicyclic components, and at most one other component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes $\half n$, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the evolution, approaches $5π/18\approx0.8727$. A ``uniform'' model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substantially simpler than the analogous formulas for the classical random graphs of Erdős and Rényi. The notions of ``excess'' and ``deficiency,'' which are significant characteristics of the generating function as well as of the graphs themselves, lead to a mathematically attractive structural theory for the uniform model. A general approach to the study of stopping configurations makes it possible to sharpen previously obtained estimates in a uniform manner and often to obtain closed forms for the constants of interest. Empirical results are presented to complement the analysis, indicating the typical behavior when $n$ is near 20000.
We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and its orientation. For this model the critical point for the emergence of a giant component is determined via a branching process approach.
The $(k_1,k_2)$-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least $k_1$ and $k_2$ respectively. For $\max\{k_1, k_2\} \geq 2$, we establish existence of the threshold edge-density $c^*=c^*(k_1,k_2)$, such that the random digraph $D(n,m)$, on the vertex set $[n]$ with $m$ edges, asymptotically almost surely has a giant $(k_1,k_2)$-core if $m/n> c^*$, and has no $(k_1,k_2)$-core if $m/n<c^*$. Specifically, denoting $\text{P}(\text{Poisson}(z)\ge k)$ by $p_k(z)$, we prove that $c^*=\min\limits_{z_1,z_2}\max\left\{\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}(…
In this paper we show the existence of a sharp threshold for the appearance of a giant component after percolation of Cartesian products of graphs under assumptions on their maximum degrees and their isoperimetric constants. In particular, this generalises a work of Ajtai, Komlós and Szemerédi from 1982 concerning percolation of the hypercube in high dimension.
Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.
Let d \geq d_0 be a sufficiently large constant. A (n,d,c \sqrt{d}) graph G is a d-regular graph over n vertices whose second largest (in absolute value) eigenvalue is at most c \sqrt{d}. For any 0 < p < 1, G_p is the graph induced by retaining each edge of G with probability p. It is known that for p > \frac{1}{d} the graph G_p almost surely contains a unique giant component (a connected component with linear number vertices). We show that for p \geq frac{5c}{\sqrt{d}} the giant component of G_p almost surely has an edge expansion of at least \frac{1}{\log_2 n}.
We consider connected components in $k$-uniform hypergraphs for the following notion of connectedness: given integers $k\ge 2$ and $1\le j \le k-1$, two $j$-sets (of vertices) lie in the same $j$-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least $j$ vertices. We prove that certain collections of $j$-sets constructed during a breadth-first search process on $j$-components in a random $k$-uniform hypergraph are reasonably regularly distributed with high probability. We use this property to provide a short proof of the asymptotic siz…
The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\PGW(λ)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $λ$. Suppose that $λ^*>λ>1$. We show that $\PGW(λ^*)$ conditioned to survive forever …
We study the joint components in a random `double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices, which contains both a red and a blue spanning tree. We show that there are critical pairs of red and blue edge densities at which a joint-giant component appears. In contrast to the standard binomial graph model, the phase transition is first order: the size of the largest joint component jumps from $O(1)$ vertices to $Θ(n)$ at the critical point. We connect this phenomenon to the properties of a certain bi…
In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp's exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.
We establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\hnp$ with edge probability $p=c/\binnd$, where $(d-1)^{-1}+\eps<c<\infty$. The proof relies on a new, purely probabilistic approach, and is based on Stein's method as well as exposing the edges of $H_d(n,p)$ in several rounds.
We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy an…
In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdős-Rényi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+ε)/n$ where $ε^3 n \to\infty$ and $ε=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = λ/n$ where $λ>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace th…
The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a pre-defined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phas…
Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\min\{k_1,k_2\}\ge 2$ and $M/N>\max\{k_1,k_2\}$, $M=Θ(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists…
We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" - an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations.
Diameter and mixing time of the giant component in the percolated hypercube
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究院）；D-MATH ETH Zürich（苏黎世联邦理工学院数学系）；Institute of Statistics and Mathematical Methods in Economics, TU Wien（维也纳工业大学统计与经济数学方法研究所）；School of Computer Science, University of Sheffield（谢菲尔德大学计算机学院）
We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component bein…
We prove a law of large numbers for the order and size of the largest strongly connected component in the directed configuration model. Our result extends previous work by Cooper and Frieze.
Two models of a random digraph on $n$ vertices, $D(n,\text{Prob}(\text{arc})=p)$ and $D(n,\text{number of arcs}=m)$ are studied. In 1990, Karp for $D(n,p)$ and independently T. Łuczak for $D(n,m=cn)$ proved that for $c>1$, with probability tending to 1, there is an unique strong component of size of order $n$. Karp showed, in fact, that the giant component has likely size asymptotic to $nθ^2$, where $θ=θ(c)$ is the unique positive root of $1-θ=e^{-c θ}$. In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant stron…
A short proof of a central limit theorem for the order of the giant component and $k$-core
Institute of Science and Technology Austria（奥地利科学技术研究所）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）；Institute of Discrete Mathematics, Graz University of Technology（格拉茨工业大学离散数学研究所）；Department of Mathematics, London School of Economics（伦敦政治经济学院数学系）
Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric space elements. We here propose a model $G=G(X, f)$, in which $(X, d)$ is a metric space, $V(G)=X$, and $\mathbb{P}(u\sim v) = f(d(u, v))$, where $f$ is a decreasing function on the set of possible distances in $X$. We consider the case that $X$ is the $n\times n \times \dots\times n$ integer lattice in dimension $r$, with $d$ the $\ell_1$ metric, and $f(d) = \f…
We provide a complete description of the giant component of the Erdős-Rényi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+ε)/n$ where $ε^3 n \to \infty$ and $ε=o(1)$. Our description is particularly simple for $ε= o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 ε^3 n$ and variance $ε^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each ed…
For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $λ(n)$ for some function $λ$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D}…
On a large finite connected graph let edges $e$ become "open" at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.
We prove a moderate deviations principles for the size of the largest connected component in a random $d$-uniform hypergraph. The key tool is a version of the exploration process, that is also used to investigate the giant component of an Erdös-Rényi graph, a moderate deviations principle for the martingale associated with this exploration process, and exponential estimates.
We study the sizes of connected components according to their excesses during a random graph process built with $n$ vertices. The considered model is the continuous one defined in Janson 2000. An ${\ell}$-component is a connected component with ${\ell}$ edges more than vertices. $\ell$ is also called the \textit{excess} of such component. As our main result, we show that when $\ell$ and ${n \over \ell}$ are both large, the expected number of vertices that ever belong to an $\ell$-component is about ${12}^{1/3} {\ell}^{1/3} n^{2/3}$. We also obtain limit theorems for the number of creations of …
We study the emergence of a giant component in the configuration model subject to additional constraints on the edges. We partition a $d$-dimensional torus into a cubic lattice with a diverging number of compartments containing vertices and allow only local edges inside and between neighbouring compartments. We show that, when the number of vertices per compartment grows quickly enough, a giant component emerges under similar conditions as for the standard configuration model. Conversely, when the compartment sizes are fixed, our model might not feature a giant component even if the standard c…
We study the simple random walk on the giant component of a supercritical Erdős-Rényi random graph on $n$ vertices, in particular the so-called vacant set at level $u$, the complement of the trajectory of the random walk run up to a time proportional to $u$ and $n$. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter $u_{\star}$: For $u<u_{\star}$ the vacant set has with high probability a unique giant component of order $n$ and all other components small, of order at most $\log^{7}n$, whereas for $u>u_{\star}$ it has with high probability…
A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). W…
We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edg…
Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-Löf, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph $G(n,p)$ above the phase transition. Here we show that the same method applies to the analogous model of random $k$-uniform hypergraphs, establishing asymptotic normality throughout the (sparse) supercritical regime. Previously, asymptotic normality was known only towards the two ends of this regime.
We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process…
Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + μn^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as $μ$ goes from $-\infty$ to $\infty$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of $μ$ and provide more propert…
We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with $p$ parts generated according to a given degree sequence $n_i^{\mathbf{d}}(n)$ which denotes the number of vertices in part $i$ of the multipartite graph with degree given by the vector $\mathbf{d}$. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size…
We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\ge 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ has a giant component in the whole supercritical phase, that is for all $h<h_\star$, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [AC20b], where it was shown that…
The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the…
Preferential attachment networks with power law exponent $τ>3$ are known to exhibit a phase transition. There is a value $ρ_{\rm c}>0$ such that, for small edge densities $ρ\leq ρ_c$ every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities $ρ>ρ_c$ there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like $\exp(-c/ \sqrt{ρ-ρ_…
We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to the work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken only within $\textbf{C}_n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constan…
We consider random walk on a discrete torus E of side-length N, in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uN^d. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const log N. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites N^d of E, and any point of E lies within distance a…
We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the con…
In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence of degree sequences converges to a probability distribution $D$, then the size of the largest component in corresponding $n$-vertex random graph is asymptotically $ρ(D)n$, where $ρ(D)$ is a constant defined by the solution to certain equations that can be interpreted as the survival probability of a branching process associated to $D$. There have been a number of papers strengthening this result in various ways; here we prove a strong form of the result (with exponential bounds on the probability of large deviations…
If we pick $n$ random points uniformly in $[0,1]^d$ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability. We prove that in $[0,1]^d$ it suffices to connect every point to $ c_{d,1} \log{\log{n}}$ points chosen randomly among its $ c_{d,2} \log{n}-$nearest neighbors to ensure a giant component of size $n - o(n)$ with high probability. This construction yields a much sparser random graph with $\sim n \log\log{n}$ instead of $\sim n \log{n}$ edges that has comparable connectivity properties. This result has …
Consider a random directed graph on $n$ vertices with independent identically distributed outdegrees with distribution $F$ having mean $μ$, and destinations of arcs selected uniformly at random. We show that if $μ>1$ then for large $n$ there is very likely to be a unique giant strong component with proportionate size given as the product of two branching process survival probabilities, one with offspring distribution $F$ and the other with Poisson offspring distribution with mean $μ$. If $μ\leq 1$ there is very likely to be no giant strong component. We also extend this to allow for $F$ varyin…
In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. …
Random K-out graphs are receiving attention as a model to construct sparse yet well-connected topologies in distributed systems including sensor networks, federated learning, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes differ in resources and requirements, inhomogeneous random K-out graphs, denoted by $H(n;μ,K_n)$, were proposed recently. Motivated by practical settings where establishing links is costly and only a bounded choice of $K_n$ is feasible ($K_n = O(1)$), we study the size of the largest connected sub-network of …
