On Vertex, Edge, and Vertex-Edge Random Graphs

Beer, Elizabeth; Fill, James Allen; Janson, Svante; Scheinerman, Edward R.

doi:10.37236/597

Cited by 6 publications

(14 citation statements)

References 17 publications

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“…Random (s-)intersection graphs have been used to model secure wireless sensor networks [12][13][14][15][16][17][18][19][20][21][22], wireless frequency hopping [22], epidemics in human populations [1,2], small-world networks [39], trust networks [40,41], social networks [2][3][4][5][6] such as collaboration networks [2][3][4] and common-interest networks [5,6]. Random intersection graphs also motivated Beer et al [42,43] to introduce a general concept of vertex random graphs that subsumes any graph model where random features are assigned to vertices, and edges are drawn based on deterministic relations between the features of the vertices.…”

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confidence: 99%

On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs

Zhao

Yağan

Gligor

2014

2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Random s-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random s-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least s items (the graph is called a random intersection graph when s = 1). A special case of particular interest is a uniform random s-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random s-intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random sintersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random s-intersection graphs).In this paper, for binomial/uniform random sintersection graphs, we present results related to kconnectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zeroone laws for the following three properties: (i) k-vertexconnectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k.

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confidence: 99%

On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs

Zhao

Yağan

Gligor

2014

2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…So, h being symmetric is a crucial condition for the proof. If φ is given to be symmetric, one can take h = φ and obtain φ = p a µ 2 -a.s. (as in the proof of Theorem 4.2. in Beer et al (2011)). However, in our case, φ is not supposed to be symmetric.…”

Section: Inclusion/exclusion Relations Between Ards Vrds and Vardsmentioning

confidence: 98%

“…Suppose that an ARD, D(n, p a ), with n ≥ 4 is represented as a VARD, D(n, Ω, µ, φ). For the proof of the theorem, we borrow some tools from functional analysis which are presented in the proof of the Theorem 4.2. in Beer et al (2011). Let h : Ω × Ω → [0, 1] be a symmetric measurable function and T be the integral operator with kernel h on the space L 2 (Ω, µ) of µ-square-integrable functions on Ω:…”

Section: Inclusion/exclusion Relations Between Ards Vrds and Vardsmentioning

confidence: 99%

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A Classification of Isomorphism-Invariant Random Digraphs

Bahadır,

Ceyhan

2016

Preprint

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We classify isomorphism-invariant random digraphs according to where randomness resides, namely, arcs, vertices, and vertices and arcs together which in turn yield arc random digraphs (ARD), vertex random digraphs (VRD) and vertex-arc random digraphs (VARD), respectively. This digraph classification can be viewed as an extension of the classification of isomorphisminvariant random graphs. We introduce randomness in the direction of the edges of a given graph and obtain direction random digraphs (DRD) as well. We classify DRDs according to which component is random in addition to the direction and study the relations of DRDs with VARDs, VRDs and ARDs. We also consider random nearest neighbor digraphs and determine their membership with respect to these digraph families.

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“…Our PICDs are random digraphs (according to the digraph version of classification of Beer et al (2011)) in which each vertex corresponds to a data point and arcs are defined in terms of some bivariate relation on the data, and are also related to the class cover catch digraph (CCCD) introduced by Priebe et al (2001) who derived the exact distribution of its domination number for uniform data from two classes in R. A CCCD consists of a vertex set in R d and arcs (u, v) if v is inside the ball centered at u with a radius based on spatial proximity of the points. CCCDs were also extended to higher dimensions and were demonstrated to be a competitive alternative to the existing methods in classification (see DeVinney and Priebe (2006) and references therein) and to be robust to the class imbalance problem (Manukyan and Ceyhan (2016)).…”

Section: Introductionmentioning

confidence: 99%

Domination number of an interval catch digraph family and its use for testing uniformity

Ceyhan

2020

Statistics

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We consider a special type of interval catch digraph (ICD) family for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with an expansion and a centrality parameter, hence we will refer to this ICD as parameterized ICD (PICD). We derive the exact (and asymptotic) distribution of the domination number of this PICD family when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire range of the parameters; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter combinations. We use the domination number for testing uniformity of data in real line, prove its consistency against certain alternatives, and compare it with two commonly used tests and three recently proposed tests in literature and also arc density of this ICD and of another ICD family in terms of size and power. Based on our extensive Monte Carlo simulations, we demonstrate that domination number of our PICD has higher power for certain types of deviations from uniformity compared to other tests.

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On Vertex, Edge, and Vertex-Edge Random Graphs

Cited by 6 publications

References 17 publications

On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs

On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs

A Classification of Isomorphism-Invariant Random Digraphs

Domination number of an interval catch digraph family and its use for testing uniformity

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