Structure of large random hypergraphs

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

Hypergraphs, the generalization of graphs in which edges become conglomerates of r nodes called hyperedges of rank r ≥ 2, are excellent models to study systems with interactions that are beyond the pairwise level. For hypergraphs, the node degree ℓ (number of hyperedges connected to a node) and the number of neighbors k of a node differ from each other in contrast to the case of graphs, where counting the number of edges is equivalent to counting the number of neighbors. In this article, I calculate the distribution of the number of node neighbors in random hypergraphs in which hyperedges of uniform rank r have a homogeneous (equal for all hyperedges) probability p to appear. This distribution is equivalent to the degree distribution of ensembles of graphs created as projections of hypergraph or bipartite network ensembles, where the projection connects any two nodes in the projected graph when they are also connected in the hypergraph or bipartite network. The calculation is non-trivial due to the possibility that neighbor nodes belong simultaneously to multiple hyperedges (node overlaps). From the exact results, the traditional asymptotic approximation to the distribution in the sparse regime (small p) where overlaps are ignored is rederived and improved; the approximation exhibits Poisson-like behavior accompanied by strong fluctuations modulated by power-law decays in the system size N with decay exponents equal to the minimum number of overlapping nodes possible for a given number of neighbors. It is shown that the dense limit cannot be explained if overlaps are ignored, and the correct asymptotic distribution is provided. The neighbor distribution requires the calculation of a new combinatorial coefficient Q r−1 (k, ℓ), which counts the number of distinct labelled hypergraphs of k nodes, ℓ hyperedges of rank r − 1, and where every node is connected to at least one hyperedge. Some identities of Q r−1 (k, ℓ) are derived and applied to the verification of normalization and the calculation of moments of the neighbor distribution.

Physical Review E, 2009

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.

2014

We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random k- We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of . 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 behavior 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 H k (n, p) has size Θ(ln(pD)N D -1 ), provided that k ≫ √ n ln n. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in H k (n, p), for essentially all values of p and k.

Combinatorics, Probability and Computing, 2017

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$ k (n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$ . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size $$(1+o(1))p\ffrac kn N$$ for any $$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ 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 $\mathcal{H}$ k (n, p) has size Θ(ln(pD)ND −1), provided that $k \gg \sqrt{n \ln n}$ . Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$ k , for essentially all values of p...

The Electronic Journal of Combinatorics

Let $c$ be a positive constant. We show that if $r=\lfloor{cn^{1/3}}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.

The Electronic Journal of Combinatorics, 2012

We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.

SIAM Journal on Discrete Mathematics, 2016

A hypergraph is k-irregular if there is no set of k vertices all of which have the same degree. We asymptotically determine the probability that a random uniform hypergraph is k-irregular.

Random Structures & Algorithms, 2011

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs.Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (Kk,M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

2007

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate the probability that a binomial random hypergraph H d (n, p) is connected, and determine the expected number of edges of H d (n, p) conditioned on its being connected. This generalizes prior work of Bender, Canfield, and McKay [2] on the number of connected graphs; however, our approach relies on elementary probabilistic methods, extending an approach of O'Connell, rather than using powerful tools from enumerative combinatorics. We also estimate the probability for each t that, given k = O(n) balls in n bins, every bin is occupied by at least t balls.