Random Graph Gn Research Articles

On the first and second largest components in the percolated random geometric graph

The percolated random geometric graph Gn(λ,p) has vertex set given by a Poisson Point Process in the square [0,n]2, and every pair of vertices at distance at most 1 independently forms an edge with probability p. For a fixed p, Penrose proved that there is a critical intensity λc=λc(p) for the existence of a giant component in Gn(λ,p). Our main result shows that for λ>λc, the size of the second-largest component is a.a.s. of order (logn)2. Moreover, we prove that the size of the largest component rescaled by n converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of G(λ,p) (which is the infinite volume version of Gn(λ,p)). Moreover, we prove that for a large class of graphs converging in a suitable sense to G(λ,1), the corresponding critical percolation thresholds converge as well to the ones of G(λ,1).

Upper tails via high moments and entropic stability

Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,p. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in Gn,p for all p=p(n) satisfying n−1logn≪p≪1.

A new proof of the KŁR conjecture

Estimating the probability that the Erdős-Rényi random graph Gn,m is H-free, for a fixed graph H, is one of the fundamental problems in random graph theory. If H is non-bipartite and m is such that each edge of Gn,m belongs to a copy of H′ for every H′⊆H, in expectation, then it is known that Gn,m is H-free with probability exp⁡(−Θ(m)). The KŁR conjecture, slightly rephrased, states that if we further condition on uniform edge distribution, the archetypal property of random graphs, the probability of being H-free becomes superexponentially small in the number of edges. While being interesting on its own, the conjecture has received significant attention due to its connection with the sparse regularity lemma, and the many results in random graphs that follow. It was proven by Balogh, Morris, and Samotij and, independently, by Saxton and Thomason, as one of the first applications of the hypergraph containers method. We give a new direct proof using induction.

Spanners in randomly weighted graphs: Independent edge lengths

Given a connected graph G=(V,E) and a length function ℓ:E→R we let dv,w denote the shortest distance between vertex v and vertex w. A t-spanner is a subset E′⊆E such that if dv,w′ denotes shortest distances in the subgraph G′=(V,E′) then dv,w′≤tdv,w for all v,w∈V. We show that for a large class of graphs with suitable degree and expansion properties with independent exponential mean one edge lengths, there is w.h.p. a 1-spanner that uses ≈12nlogn edges and that this is best possible. In particular, our result applies to the random graphs Gn,p for np≫logn.

The Largest Component of Near-Critical Random Intersection Graph with Tunable Clustering

In this paper, we study the largest component of the near-critical random intersection graph G n , m , p with n nodes and m elements, where m = Θ n which leads to the fact that the clustering is tunable. We prove that with high probability the size of the largest component in the weakly supercritical random intersection graph with tunable clustering on n vertices is of order n ϵ n , and it is of order ϵ − 2 n log n ϵ 3 n in the weakly subcritical one, where ϵ n ⟶ 0 and n 1 / 3 ϵ n ⟶ ∞ as n ⟶ ∞ .

Minimum spanning trees of random geometric graphs with location dependent weights

Consider n nodes {Xi}1≤i≤n independently distributed in the unit square S, each according to a distribution f. Nodes Xi and Xj are joined by an edge if the Euclidean distance d(Xi,Xj) is less than rn, the adjacency distance and the resulting random graph Gn is called a random geometric graph (RGG). We now assign a location dependent weight to each edge of Gn and define MSTn to be the sum of the weights of the minimum spanning trees of all components of Gn. For values of rn above the connectivity regime, we obtain upper and lower bound deviation estimates for MSTn and L2-convergence of MSTn appropriately scaled and centred.

Triangle resilience of the square of a Hamilton cycle in random graphs

Since first introduced by Sudakov and Vu in 2008, the study of resilience problems in random graphs received a lot of attention in probabilistic combinatorics. Of particular interest are resilience problems of spanning structures. It is known that for spanning structures which contain many triangles, local resilience cannot prevent an adversary from destroying all copies of the structure by removing a negligible amount of edges incident to every vertex. In this paper we generalise the notion of local resilience to H-resilience and demonstrate its usefulness on the containment problem of the square of a Hamilton cycle. In particular, we show that there exists a constant C>0 such that if p⩾Clog3⁡n/n then w.h.p. in every subgraph G of a random graph Gn,p there exists the square of a Hamilton cycle, provided that every vertex of G remains on at least a (4/9+o(1))-fraction of its triangles from Gn,p. The constant 4/9 is optimal and the value of p slightly improves on the best-known appearance threshold of such a structure and is optimal up to the logarithmic factor.

The threshold for the square of a Hamilton cycle

Resolving a conjecture of Kühn and Osthus from 2012, we show that p = 1 / n p= 1/\sqrt {n} is the threshold for the random graph G n , p G_{n,p} to contain the square of a Hamilton cycle.

Spectral edge in sparse random graphs: Upper and lower tail large deviations

In this paper, we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdős–Rényi random graph Gn,p, in the regime of p where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in mean-field approximations of related problems, this paper provides a comprehensive treatment of the large deviations of the spectral edge in this entire regime, which notably includes the well-studied case of constant average degree. In particular, for r≥1 fixed, we pin down the asymptotic probability that the top r eigenvalues are jointly greater/less than their typical values by multiplicative factors bigger/smaller than 1, in the regime mentioned above. The proof for the upper tail relies on a novel structure theorem, obtained by building on estimates in (Combin. Probab. Comput. 12 (2003) 61–72), followed by an iterative cycle removal process, which shows, conditional on the upper tail large deviation event, with high probability the graph admits a decomposition in to a disjoint union of stars and a spectrally negligible part. On the other hand, the key ingredient in the proof of the lower tail is a Ramsey-type result which shows that if the Kth largest degree of a graph is not atypically small (for some large K depending on r), then either the top eigenvalue or the rth largest eigenvalue is larger than that allowed by the lower tail event on the top r eigenvalues, thus forcing a contradiction. The above arguments reduce the problems to developing a large deviation theory for the extremal degrees which could be of independent interest.

Upper tail for homomorphism counts in constrained sparse random graphs

AbstractConsider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs (extending the known results for ), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

A scaling limit for the length of the longest cycle in a sparse random graph

We discuss the length Lc,n of the longest cycle in a sparse random graph Gn,p, p=c/n, c constant. We show that for large c there exists a function f(c) such that Lc,n/n→f(c) a.s. The function f(c)=1−∑k=1∞pk(c)e−kc where pk(c) is a polynomial in c. We are only able to explicitly give the values p1,p2, although we could in principle compute any pk. We see immediately that the length of the longest path is also asymptotic to f(c)n w.h.p.

Ramsey games near the critical threshold

A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if , then the random graph Gn, p is a.a.s. H‐Ramsey, that is, any 2‐coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0‐statement also holds, that is, there exists c > 0 such that whenever the random graph Gn, p is a.a.s. not H‐Ramsey. We show that near this threshold, even when Gn, p is not H‐Ramsey, it is often extremely close to being H‐Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2‐balanced graph H, if , then the random graph Gn, p a.a.s. has the property that every 2‐edge‐coloring without monochromatic copies of H cannot be extended to an H‐free coloring after extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, Rödl, Ruciński, and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three‐color case and show that these theorems need not hold when H is not strictly 2‐balanced.

Modularity of Erdős‐Rényi random graphs

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 ≤ 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 Gn,p with n vertices and edge‐probability p. Two key findings are that the modularity is 1+o(1) with high probability (whp) for np up to 1+o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)−1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.

On existentially complete triangle-free graphs

For a positive integer k, we say that a graph is k-existentially complete if for every 0 ⩽ a ⩽ k, and every tuple of distinct vertices x1, …, xa, y1, …, yk−a, there exists a vertex z that is joined to all of the vertices x1, …, xa and to none of the vertices y1, …, yk−a. While it is easy to show that the binomial random graph Gn,1/2 satisfies this property (with high probability) for k = (1 − o(1)) log2n, little is known about the “triangle-free” version of this problem: does there exist a finite triangle-free graph G with a similar “extension property”? This question was first raised by Cherlin in 1993 and remains open even in the case k = 4. We show that there are no k-existentially complete triangle-free graphs on n vertices with $$k\, > \,{{8\log \,n} \over {\log \,\log n}}$$, for n sufficiently large.

Opinion forming in Erdős–Rényi random graph and expanders

Assume for a graph G=V,E and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called the majority model, on the Erdős–Rényi random graph Gn,p and regular expanders. First we consider the behavior of the majority model on Gn,p with an initial random configuration, where each node is blue independently with probability pb and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely logn∕n. Furthermore, we say a graph G is λ-expander if the second-largest absolute eigenvalue of its adjacency matrix is λ. We prove that for a Δ-regular λ-expander graph if λ∕Δ is sufficiently small, then the majority model by starting from 1∕2−δn blue nodes (for an arbitrarily small constant δ>0) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an “efficient” and “fast” density classifier on regular expanders. As a by-product of our results, we show that regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node Δ-regular Ramanujan graph if the initial number of blue nodes is s≤βn, the number of blue nodes in the next round is at most cs∕Δ for some constants c,β>0.

Size of the Largest Component in a Critical Graph

In this paper, by the branching process and the martingale method, we prove that the size of the largest component in the critical random intersection graphGn,n5/3,pis asymptotically of ordern2/3and the width of scaling window isn−1/3.

Disproof of a packing conjecture of Alon and Spencer

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph Gn,1/2 typically admits a covering of a constant fraction of its edges by edge‐disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for and A1,…,At chosen uniformly and independently from the k‐subsets of {1,…,n}, what can one say about Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.

A counterexample to the DeMarco‐Kahn upper tail conjecture

Given a fixed graph H, what is the (exponentially small) probability that the number XH of copies of H in the binomial random graph Gn,p is at least twice its mean? Studied intensively since the mid 1990s, this so‐called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of for fixed ε > 0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.

Local resilience of an almost spanning k‐cycle in random graphs

The famous Pósa‐Seymour conjecture, confirmed in 1998 by Komlós, Sárközy, and Szemerédi, states that for any k ≥ 2, every graph on n vertices with minimum degree kn/(k + 1) contains the kth power of a Hamilton cycle. We extend this result to a sparse random setting. We show that for every k ≥ 2 there exists C > 0 such that if then w.h.p. every subgraph of a random graph Gn,p with minimum degree at least (k/(k + 1) + o(1))np, contains the kth power of a cycle on at least (1 − o(1))n vertices, improving upon the recent results of Noever and Steger for k = 2, as well as Allen, Böttcher, Ehrenmüller, and Taraz for k ≥ 3. Our result is almost best possible in three ways: for p ≪ n−1/k the random graph Gn,p w.h.p. does not contain the kth power of any long cycle; there exist subgraphs of Gn,p with minimum degree (k/(k + 1) + o(1))np and Ω(p−2) vertices not belonging to triangles; there exist subgraphs of Gn,p with minimum degree (k/(k + 1) − o(1))np which do not contain the kth power of a cycle on (1 − o(1))n vertices.

A large deviation approach to super-critical bootstrap percolation on the random graph [formula omitted

We consider the Erdös–Rényi random graph Gn,p and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size An∗ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables {n−An∗f(n)}n≥1 with explicit rate functions and allowing the scaling function f to vary in the widest possible range.