scholarly journals The Random Connection Model on the Torus

2014 ◽  
Vol 23 (5) ◽  
pp. 796-804
Author(s):  
LUC DEVROYE ◽  
NICOLAS FRAIMAN
Keyword(s):  
Wireless Networks ◽  
Random Graphs ◽  
High Probability ◽  
Random Connection Model

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two pointsxandyare connected with probabilityg(y−x), wheregis a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

Stochastic geometry and random graphs for the analysis and design of wireless networks

2009 ◽  
Vol 27 (7) ◽  
pp. 1029-1046 ◽  
Author(s):  
M. Haenggi ◽  
J.G. Andrews ◽  
F. Baccelli ◽  
O. Dousse ◽  
M. Franceschetti
Keyword(s):  
Wireless Networks ◽  
Random Graphs ◽  
Stochastic Geometry ◽  
Analysis And Design

Connectivity model of wireless networks via dependency links random graphs

2011 ◽  
Vol 58 (1) ◽  
pp. 122-141 ◽  
Author(s):  
Ke Zuo ◽  
Huaimin Wang ◽  
Quanyuan Wu ◽  
Dongmin Hu
Keyword(s):  
Wireless Networks ◽  
Random Graphs ◽  
Connectivity Model

scholarly journals A Note on Sparse Random Graphs and Cover Graphs

10.37236/1497 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tom Bohman ◽  
Alan Frieze ◽  
Miklós Ruszinkó ◽  
Lubos Thoma
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
The Other ◽  
Other Hand ◽  
Cover Graph

It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph $G_{n,p}$ with $p\le \kappa \log n/n$ for any constant $\kappa < 2/3$. On the other hand, this is not true for somewhat higher densities: If $p\ge \lambda (\log n)^3 / (n\log\log n)$ with $\lambda > 1/8$ then with high probability we need to delete more edges than one from every triangle. Our result has a natural algorithmic interpretation.

scholarly journals Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

2021 ◽  
Author(s):  
Thomas Bläsius ◽  
Philipp Fischbeck ◽  
Tobias Friedrich ◽  
Maximilian Katzmann
Keyword(s):  
Computational Complexity ◽  
Structural Properties ◽  
Random Graphs ◽  
Polynomial Time ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Vertex Cover ◽  
Optimal Solution ◽  
Np Complete

AbstractThe computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

scholarly journals Monochromatic cycle partitions in random graphs

2020 ◽  
pp. 1-17
Author(s):  
Richard Lang ◽  
Allan Lo
Keyword(s):  
Random Graphs ◽  
Complete Graph ◽  
High Probability ◽  
Vertex Disjoint

Abstract Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph K n can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.

Cooperative Transmission against Impersonation Attack and Authentication Error in Two-Hop Wireless Networks

2015 ◽  
Vol 9 (2) ◽  
pp. 31-59
Author(s):  
Weidong Yang ◽  
Liming Sun ◽  
Zhenqiang Xu
Keyword(s):  
Wireless Networks ◽  
High Probability ◽  
Cooperative Transmission ◽  
Impersonation Attack ◽  
Dominant Factor ◽  
Relay Nodes ◽  
Information Theoretic ◽  
Information Theoretic Security

The wireless information-theoretic security from inter-session interference has attracted considerable attention recently. A prerequisite for available works is the precise distinction between legitimate nodes and eavesdroppers. However, the authentication error always exists in the node authentication process in Two-Hop wireless networks. This paper presents an eavesdropper model with authentication error and two eavesdropping ways. Then, the number of eavesdroppers can be tolerated is analyzed while the desired secrecy is achieved with high probability in the limit of a large number of relay nodes. Final, we draw two conclusions for authentication error: 1) the impersonate nodes are chosen as relay is the dominant factor of the transmitted message leakage, and the impersonation attack does seriously decrease the number of eavesdroppers can be tolerated. 2) The error authentication to legitimate nodes is almost no effect on the number of eavesdroppers can be tolerated.

scholarly journals Karp–Sipser on Random Graphs with a Fixed Degree Sequence

2011 ◽  
Vol 20 (5) ◽  
pp. 721-741 ◽  
Author(s):  
TOM BOHMAN ◽  
ALAN FRIEZE
Keyword(s):  
Random Graphs ◽  
High Probability ◽  
Degree Sequence ◽  
Fixed Degree

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

CONNECTIVITY PROPERTIES IN RANDOM REGULAR GRAPHS WITH EDGE FAULTS

2000 ◽  
Vol 11 (02) ◽  
pp. 247-262 ◽  
Author(s):  
S. NIKOLETSEAS ◽  
K. PALEM ◽  
P. SPIRAKIS ◽  
M. YUNG
Keyword(s):  
Communication Network ◽  
Random Graphs ◽  
High Probability ◽  
Small Diameter ◽  
Regular Graphs ◽  
Connected Component ◽  
New Model

We introduce a new model of random graphs, that of random regular graphs with edge faults (which we denote by [Formula: see text]), obtained by selecting the edges of a random member of the set of all regular graphs of degree r independently and with probability p. We can thus represent a communication network in which the links fail independently and with probability f =1-p. In order to deal with this new model, we extend the notion of configurations and the translation lemma between configurations and random regular graphs provided by B. Bollobás, by introducing the concept of random configurations, to account for edge faults, and by providing an extended translation lemma between random configurations and [Formula: see text] graphs. We investigate important connectivity properties of [Formula: see text] by estimating the ranges of r, f for which, with high probability, [Formula: see text] graphs a) are highly connected b) become disconnected and c) admit a giant connected component of small diameter.

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