On Random Intersection Graphs: The Subgraph Problem

Karoński, Michał; Scheinerman, Edward R.; Singer-Cohen, Karen B.

doi:10.1017/cbo9780511566059.009

Cited by 74 publications

(133 citation statements)

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“…Notice that k 2 ∼ ln n/(2 ln ln n), therefore k 2 ≤ k 1 for large n. By the above considerations, Lemma 11,(16) and the fact that k 2 ≤ k 1 , for large n, with probability 1 − o(1/n), the following steps of the BFS process may be approximated by the branching process B − with g − i 0 ≥ b i 0 . By Lemma 3 and for given v ∈ V * with probability…”

Section: Proof Of Theoremmentioning

confidence: 86%

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Diameter, connectivity, and phase transition of the uniform random intersection graph

Rybarczyk

2011

Discrete Mathematics

121

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a b s t r a c tWe study properties of the uniform random intersection graph model G (n, m, d). We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of d, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks.

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Section: Proof Of Theoremmentioning

confidence: 86%

“…Random intersection graphs were introduced in the article of Karoński et al [16] and in Singer-Cohen's Ph.D. Thesis [23] and then followed by many other papers (for instance [1,10,11,17,21,22,24]). These works focus on random intersection graphs in which D(v i ) is assigned to v i according to the binomial distribution, i.e.…”

Section: Introductionmentioning

confidence: 99%

Diameter, connectivity, and phase transition of the uniform random intersection graph

Rybarczyk

2011

Discrete Mathematics

121

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“…However, the model adopted here differs from the random graphs discussed by Singer-Cohen et al in [11,14] where each node is assigned a key ring, one key at a time according to a Bernoulli-like mechanism (so that each key ring has a random size and has positive probability of being empty).…”

Section: Random Key Graphsmentioning

confidence: 99%

On the random graph induced by a random key predistribution scheme under full visibility

Yağan

Makowski

2008

2008 IEEE International Symposium on Information Theory

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We consider the random graph induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. We show the existence of a zero-one law for the absence of isolated nodes, and complement it by a Poisson convergence for the number of isolated nodes. Leveraging earlier results and analogies with Erdős-Renyi graphs, we explore similar results for the property of graph connectivity.

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“…We nontrivially extend the G n,m,p model (''random intersection graphs") introduced by Karoński, Sheinerman and Singer-Cohen [10] and Singer-Cohen [20]. Also, Godehardt and Jaworski [9] considered similar models.…”

Section: Introductionmentioning

confidence: 99%

“…We note the following: Note 1: When p 1 = p 2 = · · · = p m = p the general random intersection graph G n,m, p reduces to the G n,m,p as in [10] and we call it the uniform random intersection graph.…”

Section: Introductionmentioning

confidence: 99%

Large independent sets in general random intersection graphs

Nikoletseas¹,

Raptopoulos²,

Spirakis³

2008

Theoretical Computer Science

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a b s t r a c tWe investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (G n,m, p ) which includes the model of [M.Karoński, E.R. Scheinerman, K.B. Singer-Cohen, On random intersection graphs: The subgraph problem, Combinatorics, Probability and Computing journal 8 (1999), 131-159] (the ''uniform'' random intersection graph models) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs.Our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters.

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On Random Intersection Graphs: The Subgraph Problem

Cited by 74 publications

References 0 publications

Diameter, connectivity, and phase transition of the uniform random intersection graph

Diameter, connectivity, and phase transition of the uniform random intersection graph

On the random graph induced by a random key predistribution scheme under full visibility

Large independent sets in general random intersection graphs

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