The size of the giant high‐order component in random hypergraphs

Kang

Koch

2019

Electron. J. Combin.

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We consider connected components in k-uniform hypergraphs for the following notion of connectedness: given integers k ≥ 2 and 1 ≤ j ≤ 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 breadthfirst 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 size of the giant j-component shortly after it appears.

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The proof of Theorem 2 in [7] was based on a short proof of Theorem 1 due to Bollobás and Riordan [6]. The idea is to study an exploration process modelling the growth of components and analyse this process based on a branching process approximation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Size of the Giant Component in Random Hypergraphs: a Short Proof

Kang

Koch

2019

Electron. J. Combin.

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“…We note that our proofs impose two restrictions on ε, neither of which match the critical window given by ε 3 n j → ∞, conjectured in [11].…”

Section: Concluding Remarks 71 the Critical Windowmentioning

confidence: 86%

“…When parameters are clear from the context, we use H as a shorthand for H k (n, p). The following higher-dimensional analogue of the random graph phase transition for H and j-connectedness was obtained in [11,12]. 11,12]).…”

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Subcritical random hypergraphs, high-order components, and hypertrees

2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

Fang

Giudice

et al. 2019

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One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph G(n, p), the threshold for the appearance of the unique largest component (also known as the giant component) is p g = n −1 . More precisely, when p changes from (1 − ε)p g (subcritical case) to p g and then to (1 + ε)p g (supercritical case) for ε > 0, with high probability the order of the largest component increases smoothly from O(ε −2 log(ε 3 n)) to Θ(n 2/3 ) and then to (1 ± o(1))2εn. Furthermore, in the supercritical case, with high probability the largest components except the giant component are trees of order O(ε −2 log(ε 3 n)), exhibiting a structural symmetry between the subcritical random graph and the graph obtained from the supercritical random graph by deleting its giant component.As a natural generalisation of random graphs and connectedness, we consider the binomial random k-uniform hypergraph H k (n, p) (where each k-tuple of vertices is present as a hyperedge with probability p independently) and the following notion of high-order connectedness. Given an integer 1 ≤ j ≤ k − 1, two sets of j vertices are called j-connected if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least j vertices. A j-connected component is a maximal collection of pairwise j-connected j-tuples of vertices. Recently, the threshold for the appearance of the giant j-connected component in H k (n, p) and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest j-connected components, with the help of a certain class of "hypertrees" and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant j-connected component.

Vanishing of cohomology groups of random simplicial complexes

Random Struct Algorithms

Giudice

Kang

et al. 2019

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We consider k-dimensional random simplicial complexes that are generated from the binomial random (k + 1)-uniform hypergraph by taking the downward-closure, where k ≥ 2. For each 1 ≤ j ≤ k − 1, we determine when all cohomology groups with coefficients in F 2 from dimension one up to j vanish and the zero-th cohomology group is isomorphic to F 2 . This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016].