Dense Peelable Random Uniform Hypergraphs (original) (raw)
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The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump-type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex v is identifiable in n steps if there is a hyperedge containing v all of whose other vertices are identifiable in fewer steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson(β) random hypergraph Λ on a set V of N vertices, in the limit as N → ∞. Here β is a formal power series with nonnegative coefficients β0, β1, . . . , and (Λ(A)) A⊆V are independent Poisson random variables such that Λ(A), the number of hyperedges on A, has mean N βj / N j whenever |A| = j.
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A k-uniform hypergraph H = (V, E) is called ℓ-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than ℓ edges. Let H n,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the ℓ-orientability of H n,m,k for all k ≥ 3 and ℓ ≥ 2, i.e., we determine a critical quantity c * k,ℓ such that with probability 1
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Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0 < d < 1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we prove that the correct 'spectrum' of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s ≥ 2k. Our work may be viewed as a continuation of the work of Chung and Graham, although our proof techniques are different in certain important parts.
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In the last few years we have witnessed the emergence, primarily in on-line communities, of new types of social networks that require for their representation more complex graph structures than have been employed in the past. One example is the folksonomy, a tripartite structure of users, resources, and tags -- labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database. Here we propose a mathematical model of such tripartite structures which represents them as random hypergraphs. We show that it is possible to calculate many properties of this model exactly in the limit of large network size and we compare the results against observations of a real folksonomy, that of the on-line photography web site Flickr. We show that in some cases the model matches the properties of the observed network well, while in others there are significant differences, which we find to be attributable to the practice of multiple tagging, i.e., the application by a single user of many tags to one resource, or one tag to many resources.
Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability
Combinatorics, Probability and Computing, 2017
We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph mathcalH\mathcal{H}mathcalH k (n, p). For 2⩽k(n) ⩽ n/2, let N=binomnkN=\binom{n}kN=binomnk and D=binomn−kkD=\binom{n-k}kD=binomn−kk . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of mathcalH\mathcal{H}mathcalH has size (1+o(1))p\ffrac kn Nforanyfor anyforanyp\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well. A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ⩽ (n/k)1−ϵ D −1, the largest intersecting subhypergraph of mathcalH\mathcal{H}mathcalH k (n, p) has size Θ(ln(pD)ND −1), provided that kggsqrtnlnnk \gg \sqrt{n \ln n}kggsqrtnlnn . Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in mathcalH\mathcal{H}mathcalH k , for essentially all values of p...
2014
This is an extended version of the thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission. Random graphs (and more generally hypergraphs) have been extensively studied, including their first order logic. In this work we focus on certain specific aspects of this vast theory. We consider the binomial model G^d+1(n,p) of the random (d+1)-uniform hypergraph on n vertices, where each edge is present, independently of one another, with probability p=p(n). We are particularly interested in the range p(n) ∼ C(n)/n^d, after the double jump and near connectivity. We prove several zero-one, and, more generally, convergence results and obtain combinatorial applications of some