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The size of the giant component in random hypergraphs: a short proof
[article]

2018
*
arXiv
*
pre-print

We use this property to provide

arXiv:1803.02809v1
fatcat:25mzocvld5c6tcpynkmgfgwqyy
*a**short**proof**of**the*asymptotic*size**of**the**giant*j-*component*shortly after it appears. ... We prove that certain collections*of*j-sets constructed during*a*breadth-first search process on j-*components**in**a**random*k-uniform*hypergraph*are reasonably regularly distributed with high probability ...*The**proof**of*Theorem 2*in*[7] was based on*a**short**proof**of*Theorem 1 due to Bollobás and Riordan [6] . ...##
###
The Size of the Giant Component in Random Hypergraphs: a Short Proof

2019
*
Electronic Journal of Combinatorics
*

We use this property to provide

doi:10.37236/7712
fatcat:5dakywgauzfdzdststxizee65u
*a**short**proof**of**the*asymptotic*size**of**the**giant*$j$-*component*shortly after it appears. ... process on $j$-*components**in**a**random*$k$-uniform*hypergraph*are reasonably regularly distributed with high probability. ... j = 1*in*[3, 5] , where*the**size**of**the**giant**component*was shown to tend to*a*normal distribution. ...##
###
The size of the giant component in random hypergraphs
[article]

2015
*
arXiv
*
pre-print

*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. ...

*In*this paper we build on this and determine

*the*asymptotic

*size*

*of*

*the*unique

*giant*

*component*. ... alters

*the*

*size*

*of*

*the*largest

*component*giving birth to

*the*

*giant*

*component*[6] . ...

##
###
Moderate deviations for the size of the giant component in a random hypergraph
[article]

2019
*
arXiv
*
pre-print

We prove

arXiv:1907.07834v1
fatcat:h7j4dgf2xnfq7g4gsg3bsmhmsa
*a*moderate deviations principles for*the**size**of**the*largest connected*component**in**a**random*d-uniform*hypergraph*. ...*The*key tool is*a*version*of**the*exploration process, that is also used to investigate*the**giant**component**of*an Erdös-Rényi graph,*a*moderate deviations principle for*the*martingale associated with this ... Acknowledgements Research*of**the*authors was funded by*the*Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 -390685587, Mathematics Münster ...##
###
The Size of the Giant Joint Component in a Binomial Random Double Graph

2021
*
Electronic Journal of Combinatorics
*

We study

doi:10.37236/8846
fatcat:5wuz7r6nn5cp7pcrqjgulakbsu
*the*joint*components**in**a**random*'double graph' that is obtained by superposing red and blue binomial*random*graphs on $n$~vertices. ...*In*contrast to*the*standard binomial graph model,*the*phase transition is first order:*the**size**of**the*largest joint*component*jumps from $O(1)$ vertices to $\Theta(n)$ at*the*critical point. ... Acknowledgement We thank an anonymous referee for carefully reading*the*manuscript and suggesting improvements.*the*electronic journal*of*combinatorics 28(1) (2021), #P1.33 ...##
###
Largest components in random hypergraphs
[article]

2014
*
arXiv
*
pre-print

Our

arXiv:1412.6366v1
fatcat:ck3txfp67vduzmvmch4yzpoasa
*proof*extends*the*recent*short**proof*for*the*graph case by Krivelevich and Sudakov which makes use*of**a*depth-first search to reveal*the*edges*of**a**random*graph. ... We determine that*the*existence*of**a*j-tuple-connected*component*containing Θ (n^j) j-sets*in**random*k-uniform*hypergraphs*undergoes*a*phase transition and show that*the*threshold occurs at edge probability ... Again, one might wonder when*a*j-tuple-connected*giant**component**of**size*(i.e. number*of*j-sets) Θ(n j ) emerges*in**the**random*kuniform*hypergraph*. ...##
###
On the strength of connectedness of a random hypergraph
[article]

2015
*
arXiv
*
pre-print

n + (k-1) n +c/n^d-1,

arXiv:1409.1489v2
fatcat:maysnrarfrbazetwcvqz5wg2vm
*the*probability that*the**random**hypergraph*models H_d(n, m) and H_d(n, p) are k-connected tends to e^-e^-c/(k-1)!. ... such that*the*stopping time*of*having minimal degree k is equal to*the*stopping time*of*becoming k-(vertex-)connected. ... To see this fact, consider*a**hypergraph*H on n ′ vertices without such*a*large*component*and let L 1 , . . . , L ℓ be*the*vertex sets*of**the**components**of*H*in*increasing order by their cardinalities. ...##
###
Subcritical random hypergraphs, high-order components, and hypertrees
[article]

2018
*
arXiv
*
pre-print

Recently,

arXiv:1810.08107v1
fatcat:iecyarrdwne2xpiqahkdd32pga
*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*. ... 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*... Vertex-connectedness was also studied for*a*model*of*non-uniform*random**hypergraphs*,*in*which*the*probability for*a*hyperedge*of**size*t to belong to*the**hypergraph*depends on*a*parameter ω t . ...##
###
Creation and Growth of Components in a Random Hypergraph Process
[article]

2006
*
arXiv
*
pre-print

Our results give insight about

arXiv:cs/0607059v1
fatcat:3qyajkpsebglve54vcdfijbrem
*the**size**of**giant**components*inside*the*phase transition*of**random**hypergraphs*. ... We prove that*the*expected number*of*creations*of*ℓ-*component*during*a**random**hypergraph*process tends to 1 as ℓ and b tend to ∞ with*the*total number*of*vertices n such that ℓ = o(√(n/b)). ...*In*this paper, all considered*hypergraphs*are b-uniform. We will study*the*growth*of**size*and complexity*of*connected*components**of**a**random**hypergraph*process {H(n, t)} 0≤t≤1 defined as follows. ...##
###
Randomly coloring simple hypergraphs with fewer colors
[article]

2017
*
arXiv
*
pre-print

We study

arXiv:1703.05173v2
fatcat:bdtteox5ezgwfendxqv53x4urq
*the*problem*of*constructing*a*(near) uniform*random*proper q-coloring*of**a*simple k-uniform*hypergraph*with n vertices and maximum degree Δ. ... (Proper*in*that no edge is mono-colored and simple*in*that two edges have maximum intersection*of**size*one). ... Corollary 1. 3 . 3*The*graph Γ Q contains*a**giant**component*Q 0*of**size*(1 − o(1))|Q|.2 Good and bad colorings Let X ∈ Ω be*a*coloring*of*V . ...##
###
Randomly colouring simple hypergraphs
[article]

2009
*
arXiv
*
pre-print

We study

arXiv:0901.3699v1
fatcat:yemxx5t5bvfmlaywppisimun7q
*the*problem*of*constructing*a*(near)*random*proper q-colouring*of**a*simple k-uniform*hypergraph*with n vertices and maximum degree Δ. ... (Proper*in*that no edge is mono-coloured and simple*in*that two edges have maximum intersection*of**size*one). ... Note that if Glauber can take X to Y*in*one step, then it can take Y to X*in*one step. Corollary 1*The*graph Γ Q contains*a**giant**component*Q 0*of**size*(1 − o(1))|Q|. ...##
###
Randomly coloring simple hypergraphs

2011
*
Information Processing Letters
*

We study

doi:10.1016/j.ipl.2011.06.001
fatcat:5t4zxtsdszbcncq3jh7p4hmkei
*the*problem*of*constructing*a*(near) uniform*random*proper q-coloring*of**a*simple k-uniform*hypergraph*with n vertices and maximum degree ∆. ... (Proper*in*that no edge is mono-colored and simple*in*that two edges have maximum intersection*of**size*one). ... Note that if Glauber can take X to Y*in*one step, then it can take Y to X*in*one step. Corollary 1*The*graph Γ Q contains*a**giant**component*Q 0*of**size*(1 − o(1))|Q|. ...##
###
Cores of random r-partite hypergraphs

2012
*
Information Processing Letters
*

We show that

doi:10.1016/j.ipl.2011.10.017
fatcat:vefe2hnkd5drjlac7bhwpnyzmq
*the*threshold c r,k (*in*terms*of**the*average degree*of**the*graph) for appearance*of**a*k-core*in**a**random*r-partite r-uniform*hypergraph*G r,n,m is*the*same as for*a**random*r-uniform*hypergraph*... This problem was provided without*a**proof*(but with strong experimental evidence)*in**the*analysis*of**the*algorithm presented*in*[2] . ... Acknowledgements We thank*the*partial support given by*the*Brazilian National Institute*of*Science and Technology for*the*Web (grant MCT/CNPq 573871/2008-6), Canada Research Chairs Program and NSERC (Nicholas ...##
###
On the connectivity threshold for colorings of random graphs and hypergraphs
[article]

2018
*
arXiv
*
pre-print

Let Ω_q=Ω_q(H) denote

arXiv:1803.05246v2
fatcat:3d7egge5anb3nfcpggemutzzqa
*the*set*of*proper [q]-colorings*of**the**hypergraph*H. Let Γ_q be*the*graph with vertex set Ω_q and an edge σ,τ} where σ,τ are colorings iff h(σ,τ)=1. ... Here h(σ,τ) is*the*Hamming distance |{v∈ V(H):σ(v)≠τ(v)}|. We show that if H=H_n,m;k, k≥ 2,*the**random*k-uniform*hypergraph*with V=[n] and m=dn/k then w.h.p. ...*The*paper uses some*of**the*ideas from [4] which showed there is*a**giant**component**in*Γ q (G n,m ), m = dn/2 w.h.p. when q ≥ cd/ log d for c > 3/2. ...##
###
Propagation Connectivity of Random Hypergraphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-15369-3_37
fatcat:chezdao3hbbsdptgvfojsjvjey
*the*concept*of*propagation connectivity on*random*3-uniform*hypergraphs*. ... We derive upper and lower bounds for*the*propagation connectivity threshold. Our*proof*is based on*a*kind*of*large deviations analysis*of**a*time-dependent*random*walk. ... with respect to*the*emergence and*size**of**the**giant**component*. ...
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