Upper Tail Bounds for Stars

Matas Šileikis; Lutz Warnke

doi:10.37236/8493

Upper Tail Bounds for Stars

The Electronic Journal of Combinatorics ◽

10.37236/8493 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Matas Šileikis ◽

Lutz Warnke

Keyword(s):

Random Graph ◽

Accepted Standard ◽

Exponential Bounds ◽

Special Case

For $r \ge 2$, let $X$ be the number of $r$-armed stars $K_{1,r}$ in the binomial random graph $G_{n,p}$. We study the upper tail ${\mathbb P}(X \ge (1+\epsilon){\mathbb E} X)$, and establish exponential bounds which are best possible up to constant factors in the exponent (for the special case of stars $K_{1,r}$ this solves a problem of Janson and Ruciński, and confirms a conjecture by DeMarco and Kahn). In contrast to the widely accepted standard for the upper tail problem, we do not restrict our attention to constant $\epsilon$, but also allow for $\epsilon \ge n^{-\alpha}$ deviations.

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A spanning bandwidth theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548321000481 ◽

2021 ◽

pp. 1-31

Author(s):

Peter Allen ◽

Julia Böttcher ◽

Julia Ehrenmüller ◽

Jakob Schnitzer ◽

Anusch Taraz

Keyword(s):

Random Graph ◽

Random Graphs ◽

Maximum Degree ◽

Minimum Degree ◽

Spanning Subgraph ◽

Vertex Graph ◽

Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

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Who is the infector? General multi-type epidemics and real-time susceptibility processes

Advances in Applied Probability ◽

10.1017/apr.2019.25 ◽

2019 ◽

Vol 51 (2) ◽

pp. 606-631

Author(s):

Tom Britton ◽

Ka Yin Leung ◽

Pieter Trapman

Keyword(s):

Real Time ◽

Random Graph ◽

Branching Processes ◽

Large Population ◽

Graph Representation ◽

Type Model ◽

Random Weights ◽

Stochastic Epidemic ◽

Special Case ◽

Large Population Limit

AbstractWe couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

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Logconcave Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/380 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 1

Author(s):

Alan Frieze ◽

Santosh Vempala ◽

Juan Vera

Keyword(s):

Combinatorial Optimization ◽

Random Graph ◽

Random Graphs ◽

Optimization Problem ◽

Unit Cube ◽

Combinatorial Optimization Problem ◽

Total Weight ◽

General Distributions ◽

Special Case ◽

Connectivity Threshold

We propose the following model of a random graph on $n$ vertices. Let $F$ be a distribution in $R_+^{n(n-1)/2}$ with a coordinate for every pair $ij$ with $1 \le i,j \le n$. Then $G_{F,p}$ is the distribution on graphs with $n$ vertices obtained by picking a random point $X$ from $F$ and defining a graph on $n$ vertices whose edges are pairs $ij$ for which $X_{ij} \le p$. The standard Erdős-Rényi model is the special case when $F$ is uniform on the $0$-$1$ unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the $X_{ij}$ are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.

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Appendix A

An Imitation-Based Approach to Modeling Homogenous Agents Societies ◽

10.4018/978-1-59140-839-0.ch016 ◽

2011 ◽

pp. 244-275

Author(s):

Goran Trajkovski

Keyword(s):

Random Graph ◽

Graph Structure ◽

Fuzzy Relations ◽

Algebraic Structures ◽

Modeling Tool ◽

Fuzzy Graphs ◽

Special Case ◽

Mathematical Graph

The first part of this appendix presents three approaches in defining the fuzzy version (generalization) of the mathematical graph structure: graphs with fuzzy vertices, graphs with fuzzy edges, and graphs with fuzzy vertices and edges. Their advantages and shortcomings are discussed briefly. Fuzzy graphs are observed in the light of fuzzy relations theory, and as a generalization of the notion of random graph. In the second part, we generalize some fuzzy algebraic structures towards not only [0, 1] valued, but lattice, poset, and relational structured valued structures. It is exciting to see how powerful a modeling tool they are, and also to see how classical results continue to hold as but a special case of the new results.

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Threshold Functions for H-factors

Combinatorics Probability Computing ◽

10.1017/s0963548300000559 ◽

1993 ◽

Vol 2 (2) ◽

pp. 137-144 ◽

Cited By ~ 12

Author(s):

Noga Alon ◽

Raphael Yuster

Keyword(s):

Random Graph ◽

Minimum Degree ◽

Threshold Function ◽

Sharp Threshold ◽

Threshold Functions ◽

Spanning Subgraph ◽

Degree Of A Vertex ◽

Vertex Disjoint ◽

Special Case

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is , where the maximum is taken over all subgraphs (V′, E′) of H with |V′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H), then n−1/a(H) is a sharp threshold function for the property that the random graph G(n, p) contains an H-factor. That is, there are two positive constants c and C so that for p(n) = cn−1/a(H) almost surely G(n, p(n)) does not have an H-factor, whereas for p(n) = Cn−1/a(H), almost surely G(n, p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdős.

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Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

Behavioral and Brain Sciences ◽

10.1017/s0140525x18001620 ◽

2018 ◽

Vol 41 ◽

Cited By ~ 2

Author(s):

Daniel Crimston ◽

Matthew J. Hornsey

Keyword(s):

General Theory ◽

Biological Markers ◽

Natural Environment ◽

Relevant Dimension ◽

Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

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Determination of the phase structure of polymerblends by electron microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100109604 ◽

1978 ◽

Vol 36 (1) ◽

pp. 492-493

Author(s):

Dr. G. Kaemof

Keyword(s):

Screw Extruder ◽

Electron Microscopic ◽

Thin Sections ◽

Single Screw Extruder ◽

Transmission Electron ◽

The Matrix ◽

Twin Screw ◽

Special Case ◽

Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

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Test Validation Without Measurement

European Journal of Psychological Assessment ◽

10.1027/1015-5759/a000253 ◽

2016 ◽

Vol 32 (3) ◽

pp. 204-214 ◽

Cited By ~ 2

Author(s):

Emilie Lacot ◽

Mohammad H. Afzali ◽

Stéphane Vautier

Keyword(s):

Test Data ◽

Test Scores ◽

Scientific Explanation ◽

Test Validation ◽

Clinical Tests ◽

Ethical Perspective ◽

Power Of Test ◽

Memory Accuracy ◽

Social Uses ◽

Special Case

Abstract. Test validation based on usual statistical analyses is paradoxical, as, from a falsificationist perspective, they do not test that test data are ordinal measurements, and, from the ethical perspective, they do not justify the use of test scores. This paper (i) proposes some basic definitions, where measurement is a special case of scientific explanation; starting from the examples of memory accuracy and suicidality as scored by two widely used clinical tests/questionnaires. Moreover, it shows (ii) how to elicit the logic of the observable test events underlying the test scores, and (iii) how the measurability of the target theoretical quantities – memory accuracy and suicidality – can and should be tested at the respondent scale as opposed to the scale of aggregates of respondents. (iv) Criterion-related validity is revisited to stress that invoking the explanative power of test data should draw attention on counterexamples instead of statistical summarization. (v) Finally, it is argued that the justification of the use of test scores in specific settings should be part of the test validation task, because, as tests specialists, psychologists are responsible for proposing their tests for social uses.

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