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

Oliver Cooley, Mihyun Kang, Christoph Koch
2018 arXiv   pre-print
We use this property to provide 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] .  ... 
arXiv:1803.02809v1 fatcat:25mzocvld5c6tcpynkmgfgwqyy

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

Oliver Cooley, Mihyun Kang, Christoph Koch
2019 Electronic Journal of Combinatorics  
We use this property to provide 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.  ... 
doi:10.37236/7712 fatcat:5dakywgauzfdzdststxizee65u

The size of the giant component in random hypergraphs [article]

Oliver Cooley, Mihyun Kang, Christoph Koch
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] .  ... 
arXiv:1501.07835v1 fatcat:j67wz2bklrdjtbpqoakd2xygoy

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

Jingjia Liu, Matthias Löwe
2019 arXiv   pre-print
We prove 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  ... 
arXiv:1907.07834v1 fatcat:h7j4dgf2xnfq7g4gsg3bsmhmsa

The Size of the Giant Joint Component in a Binomial Random Double Graph

Mark Jerrum, Tamás Makai
2021 Electronic Journal of Combinatorics  
We study 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  ... 
doi:10.37236/8846 fatcat:5wuz7r6nn5cp7pcrqjgulakbsu

Largest components in random hypergraphs [article]

Oliver Cooley, Mihyun Kang, Yury Person
2014 arXiv   pre-print
Our 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.  ... 
arXiv:1412.6366v1 fatcat:ck3txfp67vduzmvmch4yzpoasa

On the strength of connectedness of a random hypergraph [article]

Daniel Poole
2015 arXiv   pre-print
n + (k-1) n +c/n^d-1, 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.  ... 
arXiv:1409.1489v2 fatcat:maysnrarfrbazetwcvqz5wg2vm

Subcritical random hypergraphs, high-order components, and hypertrees [article]

Oliver Cooley, Wenjie Fang, Nicola Del Giudice, Mihyun Kang
2018 arXiv   pre-print
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.  ...  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 .  ... 
arXiv:1810.08107v1 fatcat:iecyarrdwne2xpiqahkdd32pga

Creation and Growth of Components in a Random Hypergraph Process [article]

Vlady Ravelomanana
2006 arXiv   pre-print
Our results give insight about 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.  ... 
arXiv:cs/0607059v1 fatcat:3qyajkpsebglve54vcdfijbrem

Randomly coloring simple hypergraphs with fewer colors [article]

Michael Anastos, Alan Frieze
2017 arXiv   pre-print
We study 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 .  ... 
arXiv:1703.05173v2 fatcat:bdtteox5ezgwfendxqv53x4urq

Randomly colouring simple hypergraphs [article]

Alan Frieze, Pall Melsted
2009 arXiv   pre-print
We study 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|.  ... 
arXiv:0901.3699v1 fatcat:yemxx5t5bvfmlaywppisimun7q

Randomly coloring simple hypergraphs

Alan Frieze, Páll Melsted
2011 Information Processing Letters  
We study 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|.  ... 
doi:10.1016/j.ipl.2011.06.001 fatcat:5t4zxtsdszbcncq3jh7p4hmkei

Cores of random r-partite hypergraphs

Fabiano C. Botelho, Nicholas Wormald, Nivio Ziviani
2012 Information Processing Letters  
We show that 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  ... 
doi:10.1016/j.ipl.2011.10.017 fatcat:vefe2hnkd5drjlac7bhwpnyzmq

On the connectivity threshold for colorings of random graphs and hypergraphs [article]

Michael Anastos, Alan Frieze
2018 arXiv   pre-print
Let Ω_q=Ω_q(H) denote 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.  ... 
arXiv:1803.05246v2 fatcat:3d7egge5anb3nfcpggemutzzqa

Propagation Connectivity of Random Hypergraphs [chapter]

Amin Coja-Oghlan, Mikael Onsjö, Osamu Watanabe
2010 Lecture Notes in Computer Science  
We study 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.  ... 
doi:10.1007/978-3-642-15369-3_37 fatcat:chezdao3hbbsdptgvfojsjvjey
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