摘要
In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, 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 - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,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.
In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, 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 - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,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.
基金
supported by Knowledge Innovation Program of the Chinese Academy of Sciences(Grant No. kjcx-yw-s7)
the National Natural Science Foundation of China(No.10831006)
