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...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.展开更多
In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage...In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.展开更多
This paper investigates consensus of flocks consisting of n autonomous agents in the plane, where each agent has the same constant moving speed v and updates its heading by the average value of the kn nearest agents f...This paper investigates consensus of flocks consisting of n autonomous agents in the plane, where each agent has the same constant moving speed v and updates its heading by the average value of the kn nearest agents from it, with vn and kn being two prescribed parameters depending on n. Such a topological interaction rule is referred to as k,-nearest-neighbors rule, which has been validated for a class of birds by biologists and verified to be robust with respect to disturbances. A theoretical analysis will be presented for this flocking model under a random framework with large population, but without imposing any a priori connectivity assumptions. We will show that the minimum number of k~ needed for consensus is of the order O（log n） in a certain sense. To be precise, there exist two constants C1 〉 C2 〉 0 such that, if k 〉 C1 logn, then the flocking mode] will achieve consensus for any initial headings with high probability, provided that the speed vn is suitably small. On the other hand, if k 〈 Ca ]ogn, then for large n, with probability 1, there exist some initial headings such that consensus cannot be achieved, regardless of the value of Vn.展开更多
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)
文摘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 the National Natural Science Foundation of China(No.71271204)Knowledge Innovation Program of the Chinese Academy of Sciences(No.kjcx-yw-s7)
文摘In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.
文摘This paper investigates consensus of flocks consisting of n autonomous agents in the plane, where each agent has the same constant moving speed v and updates its heading by the average value of the kn nearest agents from it, with vn and kn being two prescribed parameters depending on n. Such a topological interaction rule is referred to as k,-nearest-neighbors rule, which has been validated for a class of birds by biologists and verified to be robust with respect to disturbances. A theoretical analysis will be presented for this flocking model under a random framework with large population, but without imposing any a priori connectivity assumptions. We will show that the minimum number of k~ needed for consensus is of the order O（log n） in a certain sense. To be precise, there exist two constants C1 〉 C2 〉 0 such that, if k 〉 C1 logn, then the flocking mode] will achieve consensus for any initial headings with high probability, provided that the speed vn is suitably small. On the other hand, if k 〈 Ca ]ogn, then for large n, with probability 1, there exist some initial headings such that consensus cannot be achieved, regardless of the value of Vn.
