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capacity in small world nets

capacity in small world nets - 1 A Max-Flow Min-Cut Theorem...

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arXiv:0809.1379v2 [cs.IT] 30 Jan 2009 1 A Max-Flow Min-Cut Theorem with Applications in Small Worlds and Dual Radio Networks Rui A. Costa Jo˜ao Barros Abstract Intrigued by the capacity of random networks, we start by proving a max-flow min-cut theorem that is applicable to any random graph obeying a suitably defined independence-in-cut property. We then show that this property is satisfied by relevant classes, including small world topologies, which are pervasive in both man-made and natural networks, and wireless networks of dual devices, which exploit multiple radio interfaces to enhance the connectivity of the network. In both cases, we are able to apply our theorem and derive max-flow min-cut bounds for network information flow. Index Terms random graphs, capacity, small world networks, wireless networks I. I NTRODUCTION In the quest for the fundamental limits of communication networks, whose topology is typically described by graphs, the connection between the maximum information flow and the minimum cut of the network plays a singular and prominent role. In the case where the network has one or more independent sources of information but only one sink, it is known that the transmitted information behaves like water in pipes and the capacity can be obtained by classical network Rui A. Costa is with the Instituto de Telecomunicac ¸ oes and the Departamento de Ciˆencia dos Computadores da Faculdade de Ciˆencias da Universidade do Porto, Porto, Portugal; URL: http://www.dcc.fc.up.pt/ ruicosta/ . Jo˜ao Barros is with the Instituto de Telecomunicac ¸ oes and the Departamento de Engenharia Electrot´ecnica e de Computadores da Faculdade de Engenharia da Universidade do Porto, Porto, Portugal; URL: http://paginas.fe.up.pt/ jbarros/ . This work was supported by the Fundac ¸˜ao para a Ciˆencia e Tecnologia (Portuguese Foundation for Science and Technology) under grants SFRH-BD-27273-2006 and POSC/EIA/62199/2004. Parts of this work have been presented at ITW 2006 [1], NetCod 2006 [2], and SpaSWiN 2007 [3]. January 30, 2009 DRAFT
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flow methods. Specifically, the capacity of this network will then follow from the well-known Ford-Fulkerson max-flow min-cut theorem [4], which asserts that the maximal amount of a flow (provided by the network) is equal to the capacity of a minimal cut, i.e. a nontrivial partition of the graph node set V into two parts such that the sum of the capacities of the edges connecting the two parts (the cut capacity) is minimum. Provided there is only a single sink, routing offers an optimal solution for transporting messages both when they are statistically independent [5] and when they are generated by correlated sources [6]. Max-flow min-cut arguments are useful also in the case of multicast networks, in which a single source broadcasts a number of messages to a set of sinks. This network capacity problem was solved in [7], where it is shown that applying coding operations at intermediate nodes (i.e. network coding ) is necessary to achieve the max-flow/min-cut bound of a general network.
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