Random graphG Research Articles

Fast Sampling via Spectral Independence Beyond Bounded-degree Graphs

Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.

Friendly bisections of random graphs

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph G ( n , 1 / 2 ) \mathbb {G}(n,1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n − o ( n ) n-o(n) vertices have more neighbours in their own part as across. Our proof is constructive, and in the process, we develop a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.

Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)

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 ) 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 ) G(n,p) model.

Optimal threshold for a random graph to be 2-universal

For a family of graphs F \mathcal {F} , a graph G G is F \mathcal {F} -universal if G G contains every graph in F \mathcal {F} as a (not necessarily induced) subgraph. For the family of all graphs on n n vertices and of maximum degree at most two, H ( n , 2 ) \mathcal {H}(n,2) , we prove that there exists a constant C C such that for p ≥ C ( log ⁡ n n 2 ) 1 3 , p \geq C \left ( \frac {\log n}{n^2} \right )^{\frac {1}{3}}, the binomial random graph G ( n , p ) G(n,p) is typically H ( n , 2 ) \mathcal {H}(n,2) -universal. This bound is optimal up to the constant factor as illustrated in the seminal work of Johansson, Kahn, and Vu for triangle factors. Our result improves significantly on the previous best bound of p ≥ C ( log ⁡ n n ) 1 2 p \geq C \left (\frac {\log n}{n}\right )^{\frac {1}{2}} due to Kim and Lee. In fact, we prove the stronger result that for H ℓ ( n , 2 ) \mathcal {H}^{\ell }(n,2) , the family of all graphs on n n vertices, of maximum degree at most two and of girth at least ℓ \ell , G ( n , p ) G(n,p) is typically H ℓ ( n , 2 ) \mathcal H^{\ell }(n,2) -universal when p ≥ C ( log ⁡ n n ℓ − 1 ) 1 ℓ . p \geq C \left (\frac {\log n}{n^{\ell -1}}\right )^{\frac {1}{\ell }}. This result is also optimal up to the constant factor. Our results verify (in a weak form) a classical conjecture of Kahn and Kalai.

On the Number of Solutions in Random Graphk-Colouring

Letk⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of lnZk(G(n, m)), whereZk(G(n, m)) is the number ofk-colourings of the random graphG(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles inG(n, m). Our result holds for a wide range of average degrees, and forkexceeding a certain constantk0it covers all average degrees up to the so-calledcondensation phase transition.

Planting Colourings Silently

Letk⩾ 3 be a fixed integer and letZk(G) be the number ofk-colourings of the graphG. For certain values of the average degree, the random variableZk(G(n,m)) is known to be concentrated in the sense that$\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability (Achlioptas and Coja-Oghlan,Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees,$\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability foranydiverging function$\omega=\omega(n)\ra\infty$. Forkexceeding a certain constantk0this result covers all average degrees up to the so-calledcondensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing ak-colouring of the random graphG(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

The distribution of sandpile groups of random graphs

We determine the distribution of the sandpile group (or Jacobian) of the Erdős-Rényi random graph G ( n , q ) G(n,q) as n n goes to infinity. We prove the distribution converges to a specific distribution conjectured by Clancy, Leake, and Payne. This distribution is related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the “moments” of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. The methods developed to prove this result include inverse Littlewood-Offord theorems over finite rings and new techniques for studying homomorphisms of finite abelian groups with not only precise structure but also approximate versions of that structure. We then show these moments determine a unique distribution despite their p k 2 p^{k^2} -size growth. In particular, our theorems imply universality of singularity probability and ranks mod p p for symmetric integral matrices.

Conflict-Free Colouring of Graphs

We study the conflict-free chromatic number χCFof graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number ann-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graphG(n,p) and give the asymptotics forp= ω(1/n). We also show that forp≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

Random graphs and the parity quantifier

The classical zero-one law for first-order logic on random graphs says that for every first-order property φ in the theory of graphs and every p ∈ (0,1), the probability that the random graph G ( n , p ) satisfies φ approaches either 0 or 1 as n approaches infinity. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G ( n , p ) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this work, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier, FO[⌖], over G ( n , p ), thus eluding the above hurdles. Specifically, we establish the following “modular convergence law”. For every FO[⌖] sentence φ, there are two explicitly computable rational numbers a 0 , a 1 , such that for i ∈ {0,1}, as n approaches infinity, the probability that the random graph G (2 n + i , p ) satisfies φ approaches a i . Our results also extend appropriately to FO equipped with Mod q quantifiers for prime q . In the process of deriving this theorem, we explore a new question that may be of interest in its own right. Specifically, we study the joint distribution of the subgraph statistics modulo 2 of G ( n , p ): namely, the number of copies, mod 2, of a fixed number of graphs F 1 , …, F ℓ of bounded size in G ( n , p ). We first show that every FO[⌖] property φ is almost surely determined by subgraph statistics modulo 2 of the above type. Next, we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2, and we determine this limiting distribution completely. Interestingly, both these steps are based on a common technique using multivariate polynomials over finite fields and, in particular, on a new generalization of the Gowers norm. The first step is analogous to the Razborov-Smolensky method for lower bounds for AC 0 with parity gates, yet stronger in certain ways. For instance, it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every FO[⌖] sentence, which is something that is not known for AC 0 [⌖].

Expanders Are Universal for the Class of All Spanning Trees

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

Sub-Gaussian Tails for the Number of Triangles inG(n, p)

LetXbe the random variable that counts the number of triangles in the binomial random graphG(n, p). We show that for some positive constantc, the probability thatXdeviates from its expectation by at least λVar(X)1/2is at moste−cλ2, providedp=o(1), λ = ω($\sqrt{\ln n}$) and λ ≤ (n3p3+n4p5)1/6.

Thek-Core and Branching Processes

Thek-coreof a graphGis the maximal subgraph ofGhaving minimum degree at leastk. In 1996, Pittel, Spencer and Wormald found the threshold λcfor the emergence of a non-trivialk-core in the random graphG(n, λ/n), and the asymptotic size of thek-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study thek-core in a certain power-law or ‘scale-free’ graph with a parameterccontrolling the overall density of edges. For eachk≥ 3, we find the threshold value ofcat which thek-core emerges, and the fraction of vertices in thek-core whencis ϵ above the threshold. In contrast toG(n, λ/n), this fraction tends to 0 as ϵ→0.

Upper tails for subgraph counts in random graphs

LetG be a fixed graph and letX G be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofX G which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon’s result about the maximum number of copies ofG in a graph with a given number of edges. Similar bounds are proved for the random graphG(n, M) too.

The concentration of the chromatic number of random graphs

We prove that for every constant δ>0 the chromatic number of the random graphG(n, p) withp=n−1/2−δ is asymptotically almost surely concentrated in two consecutive values. This implies that for any β<1/2 and any integer valued functionr(n)≤O(nβ) there exists a functionp(n) such that the chromatic number ofG(n,p(n)) is preciselyr(n) asymptotically almost surely.

Sudden Emergence of a Giantk-Core in a Random Graph

Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdős–Rényi random graphG(n, m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork⩾3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=minλ>0 λ/πk(λ) andπk(λ)=P{Poisson(λ)⩾k−1}. In particular,c3≈3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ≈pk(λk) n. Hereλkis the minimum point ofλ/πk(λ) andpk(λk)=P{Poisson(λk)⩾k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one.

Induced trees in sparse random graphs

The random graphG(n, p) onn vertices with edge probabilityp = c/n contains an induced tree of orderα c n whereα c > 0 forc > 1. This proves a conjecture of Erdos and Palka.