scholarly journals Hamilton $\ell$-Cycles in Randomly Perturbed Hypergraphs

10.37236/7671 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Andrew McDowell ◽  
Richard Mycroft
Keyword(s):  
High Probability ◽  
Analogous Result ◽  
Uniform Hypergraph ◽  
Small Constant

We prove that for integers $2 \leqslant \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly 'perturbed' by changing non-edges to edges independently at random with probability $p \geqslant O(n^{-(k-\ell)-c})$, then with high probability the resulting $k$-uniform hypergraph contains a Hamilton $\ell$-cycle. This complements a recent analogous result for Hamilton $1$-cycles due to Krivelevich, Kwan and Sudakov, and a comparable theorem in the graph case due to Bohman, Frieze and Martin.

scholarly journals Perfect Fractional Matchings in $k$-Out Hypergraphs

10.37236/6890 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Pat Devlin ◽  
Jeff Kahn
Keyword(s):  
High Probability ◽  
Short Proof ◽  
Analogous Result ◽  
Stopping Time ◽  
Uniform Hypergraph ◽  
Time Result

Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to\infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.

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.

FAULTY RANDOM GEOMETRIC NETWORKS

2000 ◽  
Vol 10 (04) ◽  
pp. 343-357 ◽  
Author(s):  
JOSEP DIAZ ◽  
JORDI PETIT ◽  
MARIA SERNA
Keyword(s):  
Failure Probability ◽  
High Probability ◽  
Hamiltonian Cycle ◽  
Computational Power ◽  
Linear Arrangement ◽  
Node Failure ◽  
Small Constant ◽  
Distributed Computations ◽  
Order Of Magnitude ◽  
Edge Failure

In this paper we analyze the computational power of random geometric networks in the presence of random (edge or node) faults considering several important network parameters. We first analyze how to emulate an original random geometric network G on a faulty network F. Our results state that, under the presence of some natural assumptions, random geometric networks can tolerate a constant node failure probability with a constant slowdown. In the case of constant edge failure probability the slowdown is an arbitrarily small constant times the logarithm of the graph order. Then we consider several network measures, stated as linear layout problems (Bisection, Minimum Linear Arrangement and Minimum Cut Width). Our results show that random geometric networks can tolerate a constant edge (or node) failure probability while maintaining the order of magnitude of the measures considered here. Finally we show that, with high probability, random geometric networks with (edge or node) faults do have a Hamiltonian cycle, provided the failure probability is constant. Such capability enables performing distributed computations based on end-to-end communication protocols.

scholarly journals Co-degrees Resilience for Perfect Matchings in Random Hypergraphs

10.37236/8167 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Asaf Ferber ◽  
Lior Hirschfeld
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Uniform Hypergraph ◽  
Hypergraph Model ◽  
Large N ◽  
Random Hypergraphs

In this paper we prove an optimal co-degrees resilience property for the binomial $k$-uniform hypergraph model $H_{n,p}^k$ with respect to perfect matchings. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log {n}/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.

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

10.37236/7712 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Oliver Cooley ◽  
Mihyun Kang ◽  
Christoph Koch
Keyword(s):  
High Probability ◽  
Short Proof ◽  
The Other ◽  
Connected Components ◽  
Search Process ◽  
Uniform Hypergraph ◽  
Breadth First Search ◽  
Asymptotic Size ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs

We consider connected components in $k$-uniform hypergraphs for the following notion of connectedness: given integers $k\ge 2$ and $1\le j \le k-1$, two $j$-sets (of vertices) lie in the same $j$-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least $j$ vertices.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. We use this property to provide a short proof of the asymptotic size of the giant $j$-component shortly after it appears.

scholarly journals Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

10.37236/5025 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Asaf Ferber
Keyword(s):  
Random Graph ◽  
Directed Graph ◽  
Random Graphs ◽  
High Probability ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Edge Probability

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd $n$ their $p$ was $\omega((\log n)/n)$. Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we randomly color the edge set of a random directed graph $D_{n,p}$ with $(1+o(1))n$ colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.

APPROXIMATING CENTER POINTS WITH ITERATIVE RADON POINTS

1996 ◽  
Vol 06 (03) ◽  
pp. 357-377 ◽  
Author(s):  
KENNETH L. CLARKSON ◽  
DAVID EPPSTEIN ◽  
GARY L. MILLER ◽  
CARL STURTIVANT ◽  
SHANG-HUA TENG
Keyword(s):  
Monte Carlo ◽  
High Probability ◽  
Constant Factor ◽  
Monte Carlo Algorithm ◽  
High Quality ◽  
Center Point ◽  
Partitioning Methods ◽  
Small Constant ◽  
Mesh Partitioning ◽  
Point Set

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in Rd. A point c∈Rd is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d+1)-center point; our algorithm finds an Ω(1/d2)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O( log 2d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ∊-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly.

Quick Consult: Quick Consult: Acromioplasty

2002 ◽  
Vol 7 (3) ◽  
pp. 4-5
Keyword(s):  
Glasgow Coma Scale ◽  
Outcome Assessment ◽  
High Probability ◽  
Vegetative State ◽  
Behavioral Impairment ◽  
Whole Person ◽  
Ordinal Scales ◽  
Probability Of Death ◽  
The Individual ◽  
Assessment Scales

Abstract Different jurisdictions use the AMA Guides to the Evaluation of Permanent Impairment (AMA Guides) for different purposes, and this article reviews a specific jurisdictional definition in the Province of Ontario of catastrophic impairment that incorporates the AMA Guides. In Ontario, a whole person impairment (WPI) exceeding 54% or a mental or behavioral impairment of Class 4 or 5 qualifies the individual for catastrophic benefits, and individuals who do not meet the test receive a lesser benefit. By inference, this establishes a parity threshold among dissimilar injuries and dissimilar outcome assessment scales for benefits. In Ontario, the Glasgow Coma Scale (GCS) identifies patients who have a high probability of death or of severely disabled survival. The GCS recognizes gradations of vegetative state and disability, but translating the gradations for rating individual impairment on ordinal scales into a method of assessing percentage impairments cannot be done reliably, as explained in the AMA Guides, Fifth Edition. The AMA Guides also notes that mental and behavioral impairment in Class 4 (marked impairment) or 5 (extreme impairment) indicates “catastrophic impairment” by significantly impeding useful functioning (Class 4) or significantly impeding useful functioning and implying complete dependency on another person for care (Class 5). Translating the AMA Guides guidelines into ordinal scales cannot be done reliably.

The Effect of Outcome Probability on Generalization in Predictive Learning

2019 ◽  
Vol 66 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Hadar Ram ◽  
Dieter Struyf ◽  
Bram Vervliet ◽  
Gal Menahem ◽  
Nira Liberman
Keyword(s):  
High Probability ◽  
Lower Probability ◽  
Perceptual Similarity ◽  
Learning Paradigm ◽  
Predictive Learning ◽  
Lightning Bolt ◽  
Outcome Probability

Abstract. People apply what they learn from experience not only to the experienced stimuli, but also to novel stimuli. But what determines how widely people generalize what they have learned? Using a predictive learning paradigm, we examined the hypothesis that a low (vs. high) probability of an outcome following a predicting stimulus would widen generalization. In three experiments, participants learned which stimulus predicted an outcome (S+) and which stimulus did not (S−) and then indicated how much they expected the outcome after each of eight novel stimuli ranging in perceptual similarity to S+ and S−. The stimuli were rings of different sizes and the outcome was a picture of a lightning bolt. As hypothesized, a lower probability of the outcome widened generalization. That is, novel stimuli that were similar to S+ (but not to S−) produced expectations for the outcome that were as high as those associated with S+.

A Comparative Analysis of D-Dimer Assays in Patients with Clinically Suspected Pulmonary Embolism

1993 ◽  
Vol 70 (03) ◽  
pp. 408-413 ◽  
Author(s):  
Edwin J R van Beek ◽  
Bram van den Ende ◽  
René J Berckmans ◽  
Yvonne T van der Heide ◽  
Dees P M Brandjes ◽  
...  
Keyword(s):  
Pulmonary Embolism ◽  
Comparative Analysis ◽  
High Probability ◽  
Pulmonary Emboli ◽  
Plasma Samples ◽  
Suspected Pulmonary Embolism ◽  
Lung Scan ◽  
D Dimer

SummaryTo avoid angiography in patients with clinically suspected pulmonary embolism and non-diagnostic lung scan results, the use of D-dimer has been advocated. We assessed plasma samples of 151 consecutive patients with clinically suspected pulmonary embolism. Lung scan results were: normal (43), high probability (48) and non-diagnostic (60; angiography performed in 43; 12 pulmonary emboli). Reproducibility, cut-off values, specificity, and percentage of patients in whom angiography could be avoided (with sensitivity 100%) were determined for two latex and four ELISA assays.The latex methods (cut-off 500 μg/1) agreed with corresponding ELISA tests in 83% (15% normal latex, abnormal ELISA) and 81% (7% normal latex, abnormal ELISA). ELISA methods showed considerable within- (2–17%) and between-assay Variation (12–26%). Cut-off values were 25 μg/l (Behring), 50 μg/l (Agen), 300 μg/l (Stago) and 550 μg/l (Organon). Specificity was 14–38%; in 4–15% of patients angiography could be avoided.We conclude that latex D-dimer assays appear not useful, whereas ELISA methods may be of limited value in the exclusion of pulmonary embolism.

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