Minimum node degree and k-connectivity in wireless networks with unreliable links

Zhao, Jun

doi:10.1109/isit.2014.6874832

Cited by 17 publications

(9 citation statements)

References 23 publications

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“…Then k-vertexconnectivity, k-edge-connectivity, and the property of minimum vertex degree being at least k, are given by events κ v ≥ k, κ e ≥ k, and δ ≥ k, respectively. For any graph, the vertex connectivity is at most the edge connectivity, and the edge connectivity is at most the minimum vertex degree [46][47][48]. Therefore, κ v ≤ κ e ≤ δ holds.…”

Section: Graphmentioning

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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Section: Graphmentioning

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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“…al., 2014 (Ling & Tian, 2007) (Meghanathan & Gorla, 2010) ( Penrose, 1999) (Reif & Spirakis, 1985) (Xing et. al., 2009) (Zhao, 2014) ( Zhao, et. al., 2014) The aim of all of these approaches is to find a relation between the probability of having a kconnected graph and other parameters such as minimum or maximum node degrees, number of nodes and the area that the nodes are distributed.…”

Section: Probabilistic Approaches For K Detectionmentioning

confidence: 99%

On k-Connectivity Problems in Distributed Systems

Akram

Dağdeviren

2016

Advanced Methods for Complex Network Analysis

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k-Connectivity detection and restoration are important problems in graph theory and computer networks. A graph is k-connected if it remains connected after removing k-1 arbitrary nodes. The k-connectivity is an important property of a distributed system because a k-connected network can tolerate k-1 node failures without losing the network connectivity. To achieve the k-connectivity in a network, we need to determine the current connectivity value and try to increase connectivity if the current k is lower than the desired value. This chapter reviews the central and distributed algorithms for detecting and restoring the k-connectivity in graphs and distributed systems. The algorithms will be compared from complexity, accuracy and efficiency perspectives.

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“…For k ≥ 2, Wan et al [15] determine the exact formula of r n such that graph H (S) (n, r n ) or H (D) (n, r n ) is asymptotically k-connected for k ≥ 2 with a certain probability. Recently, Penrose [12] and Zhao [18] investigate connectivity in the intersection of a two-dimensional geometric random graph and an Erdős-Rényi graph.…”

Section: Related Workmentioning

confidence: 99%

The absence of isolated node in geometric random graphs

Zhao

2015

2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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One-dimensional geometric random graphs are constructed by distributing n nodes uniformly and independently on a unit interval and then assigning an undirected edge between any two nodes that have a distance at most rn. These graphs have received much interest and been used in various applications including wireless networks. A threshold of rn for connectivity is known as r * n = ln n n in the literature. In this paper, we prove that a threshold of rn for the absence of isolated node is ln n 2n (i.e., a half of the threshold r * n ). Our result shows there is a gap between thresholds of connectivity and the absence of isolated node in one-dimensional geometric random graphs; in particular, when rn equals c ln n n for a constant c ∈ ( 1 2 , 1), a one-dimensional geometric random graph has no isolated node but is not connected. This gap in onedimensional geometric random graphs is in sharp contrast to the prevalent phenomenon in many other random graphs such as two-dimensional geometric random graphs, Erdős-Rényi graphs, and random intersection graphs, all of which in the asymptotic sense become connected as soon as there is no isolated node.

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Minimum node degree and k-connectivity in wireless networks with unreliable links

Cited by 17 publications

References 23 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

On k-Connectivity Problems in Distributed Systems

The absence of isolated node in geometric random graphs

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