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ON THE hDIAMETER OF A RANDOM POINT SET
"... ABSTRACT. Let S� ∑ be a measurable space and let h: S ¢ S � R be a real symmetric Borel/∑¢∑measurable function on S ¢ S. Let B be a nonempty measurable subset in S and let µ be a probability measure supported on the restriction of the measurable space S � ∑ to B. Let B have finite hdiameter h � e ..."
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ABSTRACT. Let S� ∑ be a measurable space and let h: S ¢ S � R be a real symmetric Borel/∑¢∑measurable function on S ¢ S. Let B be a nonempty measurable subset in S and let µ be a probability measure supported on the restriction of the measurable space S � ∑ to B. Let B have finite hdiameter h � ess sup�h u�v: u�v B � � ∞� Let U�U1�U2�.... be a sequence of independent random points taking values in B according to µ and let Hn � max�h Ui�Uj:1 � i � j � n� denote the hdiameter of the set �Ui�i � 1�����n�, the maximum pairwise hdistance among the first n points. A theoretical framework is provided from which one may deduce the weak convergence of H n, upon suitable centering and rescaling, to an extremevalue distribution. The sufficient condition provided herein is quite different from that of Appel, et al. [1]. Several applications of the theory are provided.
Optimizing Mobile Networks Connectivity and Routing Using Percolation Theory and Epidemic Algorithms
"... Abstract: Mobile Adhoc NETworks (MANETs) are complex systems presenting a phase transition phenomenon: an abrupt change in the behavior of the network around a critical value of a certain key parameter. The reliability and connectivity of MANETs depend on their size and on the efficiency of the rou ..."
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Abstract: Mobile Adhoc NETworks (MANETs) are complex systems presenting a phase transition phenomenon: an abrupt change in the behavior of the network around a critical value of a certain key parameter. The reliability and connectivity of MANETs depend on their size and on the efficiency of the routing protocol. In this paper we propose a new approach in MANETs modeling, combining percolation theory and epidemic algorithms. Using percolation theory we show the existence of a connectivity threshold (in a square lattice) needed to guarantee the communications in the network, in particular when the number of direct links are limited. Epidemic algorithms are used to provide a good propagation of information in the network wile minimizing the resources cost (energy, number of messages...). 1