On the s-Colorful Number of a Random Hypergraph

2021 ◽  
Author(s):  
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.

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.

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 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 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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