On panchromatic colourings of a random hypergraph

2018 ◽  
Vol 73 (4) ◽  
pp. 731-733 ◽  
Author(s):  
D. A. Kravtsov ◽  
N. E. Krokhmal ◽  
D. A. Shabanov
Keyword(s):  
Random Hypergraph

scholarly journals Semi-Supervised Classification via Hypergraph Convolutional Extreme Learning Machine

2021 ◽  
Vol 11 (9) ◽  
pp. 3867
Author(s):  
Zhewei Liu ◽  
Zijia Zhang ◽  
Yaoming Cai ◽  
Yilin Miao ◽  
Zhikun Chen
Keyword(s):  
Extreme Learning Machine ◽  
Supervised Classification ◽  
Structured Data ◽  
Model Complex ◽  
Euclidean Domain ◽  
Noise Data ◽  
Data Points ◽  
Learning Machine ◽  
Random Hypergraph ◽  
Special Case

Extreme Learning Machine (ELM) is characterized by simplicity, generalization ability, and computational efficiency. However, previous ELMs fail to consider the inherent high-order relationship among data points, resulting in being powerless on structured data and poor robustness on noise data. This paper presents a novel semi-supervised ELM, termed Hypergraph Convolutional ELM (HGCELM), based on using hypergraph convolution to extend ELM into the non-Euclidean domain. The method inherits all the advantages from ELM, and consists of a random hypergraph convolutional layer followed by a hypergraph convolutional regression layer, enabling it to model complex intraclass variations. We show that the traditional ELM is a special case of the HGCELM model in the regular Euclidean domain. Extensive experimental results show that HGCELM remarkably outperforms eight competitive methods on 26 classification benchmarks.

On the s-Colorful Number of a Random Hypergraph

2021 ◽  
Author(s):  
D. A. Shabanov
Keyword(s):  
Random Hypergraph

scholarly journals Orientability Thresholds for Random Hypergraphs

2015 ◽  
Vol 24 (5) ◽  
pp. 774-824 ◽  
Author(s):  
PU GAO ◽  
NICHOLAS WORMALD
Keyword(s):  
Load Balancing ◽  
General Version ◽  
Line Load ◽  
Random Hypergraph ◽  
Random Hypergraphs

Leth>w> 0 be two fixed integers. LetHbe a random hypergraph whose hyperedges are all of cardinalityh. Tow-orienta hyperedge, we assign exactlywof its vertices positive signs with respect to the hyperedge, and the rest negative signs. A (w,k)-orientation ofHconsists of aw-orientation of all hyperedges ofH, such that each vertex receives at mostkpositive signs from its incident hyperedges. Whenkis large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, whenh= 2 andw= 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran. This settled a conjecture of Karp and Saks.

scholarly journals The phase transition in a random hypergraph

2002 ◽  
Vol 142 (1) ◽  
pp. 125-135 ◽  
Author(s):  
Michał Karoński ◽  
Tomasz Łuczak
Keyword(s):  
Phase Transition ◽  
Random Hypergraph

scholarly journals Threshold and Hitting Time for High-Order Connectedness in Random Hypergraphs

10.37236/5064 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Oliver Cooley ◽  
Mihyun Kang ◽  
Christoph Koch
Keyword(s):  
High Probability ◽  
Hitting Time ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Time Result ◽  
Edge Probability ◽  
The Moment ◽  
Definition Of ◽  
Random Hypergraph ◽  
Random Hypergraphs

We consider the following definition of connectedness in $k$-uniform hypergraphs: 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. The hypergraph is $j$-connected if all $j$-sets are pairwise $j$-connected. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process – the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises the classical hitting time result of Bollobás and Thomason for graphs.

scholarly journals Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

10.37236/477 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Alan Frieze
Keyword(s):  
Absolute Constant ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
Random Hypergraph

In the random hypergraph $H=H_{n,p;3}$ each possible triple appears independently with probability $p$. A loose Hamilton cycle can be described as a sequence of edges $\{x_i,y_i,x_{i+1}\}$ for $i=1,2,\ldots,n/2$ where $x_1,x_2,\ldots,x_{n/2},y_1,y_2,\ldots,y_{n/2}$ are all distinct. We prove that there exists an absolute constant $K>0$ such that if $p\geq {K\log n\over n^2}$ then $$\lim_{\textstyle{n\to \infty\atop 4|n}}\Pr(H_{n,p;3}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$

scholarly journals Asymptotic normality of the size of the giant component in a random hypergraph

2012 ◽  
Vol 41 (4) ◽  
pp. 441-450 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Asymptotic Normality ◽  
Giant Component ◽  
Random Hypergraph

On the concentration of the chromatic number of a random hypergraph

2017 ◽  
Vol 96 (1) ◽  
pp. 321-325 ◽  
Author(s):  
D. A. Shabanov
Keyword(s):  
Chromatic Number ◽  
Random Hypergraph

scholarly journals Counting Connected Hypergraphs via the Probabilistic Method

2016 ◽  
Vol 25 (1) ◽  
pp. 21-75 ◽  
Author(s):  
BÉLA BOLLOBÁS ◽  
OLIVER RIORDAN
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Local Limit Theorem ◽  
Probabilistic Method ◽  
Local Limit ◽  
Probabilistic Methods ◽  
Uniform Hypergraphs ◽  
Asymptotic Formulae ◽  
Random Hypergraph ◽  
Future Work

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on [n] = {1,2,. . .,n} withmedges, whenevernand the nullitym−n+1 tend to infinity. LetCr(n,t) be the number of connectedr-uniform hypergraphs on [n] with nullityt= (r−1)m−n+1, wheremis the number of edges. Forr≥ 3, asymptotic formulae forCr(n,t) are known only for partial ranges of the parameters: in 1997 Karoński and Łuczak gave one fort=o(logn/log logn), and recently Behrisch, Coja-Oghlan and Kang gave one fort=Θ(n). Here we prove such a formula for any fixedr≥ 3 and anyt=t(n) satisfyingt=o(n) andt→∞ asn→∞, complementing the last result. This leaves open only the caset/n→∞, which we expect to be much simpler, and will consider in future work. The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. We deduce this from the corresponding central limit theorem by smoothing techniques.

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