Largest Components in Random Hypergraphs

Cooley, Oliver; Kang, Mihyun; Person, Yury

doi:10.1017/s096354831800010x

Cited by 16 publications

(37 citation statements)

References 23 publications

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“…The process T * will be formally introduced in Section 2. It follows from the bounded degree lemma in [11] (we use a strengthening of this result: Lemma 14 in this paper) that these two couplings have essentially the same behavior. For this overview we therefore consider only T * .…”

Section: Proof Outline: Motivating Smoothnessmentioning

confidence: 84%

“…Recently, Cooley, Kang, and Person , and independently Lu and Peng , proved that for all

1 \leq j \leq k - 1

, the phase transition for the largest j ‐component in

{scriptH}^{k} (n, p)

occurs at the critical probability threshold of

{\overset{p}{true}}_{normalg} = {\overset{p}{true}}_{normalg} (n, k, j) : = \frac{1}{true(\begin{matrix} k \\ j \end{matrix} true) - 1} \frac{1}{true(\begin{matrix} n \\ k - j \end{matrix} true)} .

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The following lemma provides a crude upper bound on the set degrees in

B_{J} (t)

for general

0 \leq ℓ < j

. It is a more widely applicable strengthening of Lemma 12 from .Lemma With probability at least

1 - \exp (- Θ (n^{δ / 2}))

, for every j‐set J, every

ℓ = 0, \dots, j - 1

and every

t = 0, 1, \dots, true(\begin{matrix} n \\ k \end{matrix} true)

the process

BFS (J)

with stopping conditions (S1), (S2) , and (S3) has the following property :

Δ_{ℓ} (B_{J} (t)) = O (t p n^{- ℓ} + n^{normalδ}) .

In particular, if for some

i = 0, \dots, i_{1}

and

ℓ = 0, \dots, j - 1

we have

| C_{J} (i) | \geq n^{ℓ + δ}

, then

Δ_{ℓ} (C_{J} (i)) = O true(n^{- ℓ} | C_{J} (i) | true) .

…”

Section: Smooth Explorationmentioning

confidence: 99%

“…Such an edge contributes at most

true(\begin{matrix} k - ℓ \\ j - ℓ \end{matrix} true)

d_{L} (B_{J} (t))

. A pivot at L occurs when we query any k ‐set from a j ‐set containing L and it forms an edge. Such an edge contributes at most

true(\begin{matrix} k - ℓ \\ j - ℓ \end{matrix} true) - 1

d_{L} (B_{J} (t))

. The concepts of jumps and pivots have already been used in . A significantly more precise investigation of jumps and pivots will prove crucial in the proof of Lemma 9 in Section 4.7.Proof of Lemma Clearly the contribution of J to the degree of any

ℓ

‐set is at most one and thus negligible, so we ignore it.…”

Section: Smooth Explorationmentioning

confidence: 99%

“…For an upper bound, we also assume

d_{L}^{(jp)} (B_{J} (t)) \geq n^{normalδ}

. The following argument appeared in , but we repeat it here for completeness. We think of the branching processes as being subtrees of the infinite

c_{ℓ} N

‐ary tree, where

N = true(\begin{matrix} n \\ k - j \end{matrix} true)

and in which the children of a vertex are partitioned into clusters of size

c_{ℓ}

and the edges to such a cluster are all present with probability p , or all absent with probability

1 - p

, independently for each cluster.…”

Section: Smooth Explorationmentioning

confidence: 99%

See 4 more Smart Citations

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

Cooley

Kang

Koch

2018

Random Struct Algorithms

Self Cite

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The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any k≥3 and 1≤j≤k−1.

show abstract

Section: Proof Outline: Motivating Smoothnessmentioning

confidence: 84%

“…Recently, Cooley, Kang, and Person , and independently Lu and Peng , proved that for all

1 \leq j \leq k - 1

, the phase transition for the largest j ‐component in

{scriptH}^{k} (n, p)

occurs at the critical probability threshold of

{\overset{p}{true}}_{normalg} = {\overset{p}{true}}_{normalg} (n, k, j) : = \frac{1}{true(\begin{matrix} k \\ j \end{matrix} true) - 1} \frac{1}{true(\begin{matrix} n \\ k - j \end{matrix} true)} .

…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The following lemma provides a crude upper bound on the set degrees in

B_{J} (t)

for general

0 \leq ℓ < j

. It is a more widely applicable strengthening of Lemma 12 from .Lemma With probability at least

1 - \exp (- Θ (n^{δ / 2}))

, for every j‐set J, every

ℓ = 0, \dots, j - 1

and every

t = 0, 1, \dots, true(\begin{matrix} n \\ k \end{matrix} true)

the process

BFS (J)

with stopping conditions (S1), (S2) , and (S3) has the following property :

Δ_{ℓ} (B_{J} (t)) = O (t p n^{- ℓ} + n^{normalδ}) .

In particular, if for some

i = 0, \dots, i_{1}

and

ℓ = 0, \dots, j - 1

we have

| C_{J} (i) | \geq n^{ℓ + δ}

, then

Δ_{ℓ} (C_{J} (i)) = O true(n^{- ℓ} | C_{J} (i) | true) .

…”

Section: Smooth Explorationmentioning

confidence: 99%

“…Such an edge contributes at most

true(\begin{matrix} k - ℓ \\ j - ℓ \end{matrix} true)

d_{L} (B_{J} (t))

. A pivot at L occurs when we query any k ‐set from a j ‐set containing L and it forms an edge. Such an edge contributes at most

true(\begin{matrix} k - ℓ \\ j - ℓ \end{matrix} true) - 1

d_{L} (B_{J} (t))

ℓ

‐set is at most one and thus negligible, so we ignore it.…”

Section: Smooth Explorationmentioning

confidence: 99%

“…For an upper bound, we also assume

d_{L}^{(jp)} (B_{J} (t)) \geq n^{normalδ}

. The following argument appeared in , but we repeat it here for completeness. We think of the branching processes as being subtrees of the infinite

c_{ℓ} N

‐ary tree, where

N = true(\begin{matrix} n \\ k - j \end{matrix} true)

and in which the children of a vertex are partitioned into clusters of size

c_{ℓ}

and the edges to such a cluster are all present with probability p , or all absent with probability

1 - p

, independently for each cluster.…”

Section: Smooth Explorationmentioning

confidence: 99%

See 3 more Smart Citations

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

Cooley

Kang

Koch

2018

Random Struct Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

Vanishing of cohomology groups of random simplicial complexes

Cooley

Giudice

Kang

et al. 2019

Random Struct Algorithms

Self Cite

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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].

show abstract

Forcing large tight components in 3-graphs

Georgakopoulos

Haslegrave

Montgomery

2019

European Journal of Combinatorics

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Any n-vertex 3-graph with minimum codegree at least ⌊n/3⌋ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than ⌈2n/3⌉ vertices. Motivated by this observation, we ask which codegree forces a tight component of at least any given size. The corresponding function seems to have infinitely many discontinuities, but we provide upper and lower bounds, which asymptotically converge as the function nears the origin.

show abstract

Largest Components in Random Hypergraphs

Cited by 16 publications

References 23 publications

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

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

Vanishing of cohomology groups of random simplicial complexes

Forcing large tight components in 3-graphs

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