# Some properties for the largest component of random geometric graphs with applications in sensor networks

Some properties for the largest component of random geometric graphs with applications in sensor... In this paper we consider the standard Poisson Boolean model of random geometric graphs G( $$\mathcal{H}_{\lambda ,s}$$ ; 1) in ℝ d and study the properties of the order of the largest component L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1)). We prove that E[L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1))] is smooth with respect to λ, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs (not asymptotic graphs as s → ∞) in the high dimensional space (d ≥ 2). Moreover, we investigate the convergence rate of E[L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1))]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Some properties for the largest component of random geometric graphs with applications in sensor networks

, Volume 25 (4) – Sep 8, 2009
14 pages

/lp/springer-journals/some-properties-for-the-largest-component-of-random-geometric-graphs-lg0SzOEodB
Publisher
Springer Journals
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Math Applications in Computer Science; Applications of Mathematics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-008-8809-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper we consider the standard Poisson Boolean model of random geometric graphs G( $$\mathcal{H}_{\lambda ,s}$$ ; 1) in ℝ d and study the properties of the order of the largest component L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1)). We prove that E[L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1))] is smooth with respect to λ, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs (not asymptotic graphs as s → ∞) in the high dimensional space (d ≥ 2). Moreover, we investigate the convergence rate of E[L 1(G( $$\mathcal{H}_{\lambda ,s}$$ ; 1))]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Sep 8, 2009

### References

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