Beyond graphs: Capturing groups in networks

Ramanathan, Ram; Bar-Noy, Amotz; Basu, Prithwish; Johnson, Matthew; Ren, Wei; Swami, Ananthram; Zhao, Qing

doi:10.1109/infcomw.2011.5928935

Cited by 17 publications

(16 citation statements)

References 21 publications

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“…While a graph is given by a vertex set V and an edge set E consisting of cardinality-2 subsets of V , a hypergraph [1] is free of the constraint on the cardinality of an edge. Specifically, any non-empty subset of V can be an element (referred to as a hyperedge) of the edge set E. Hypergraphs can thus capture group behaviors and higher-dimensional relationships in complex networks that are more than a simple union of pairwise relationships [2]. In a directed hypergraph [3], each hyperedge is directed, going from a source vertex to a non-empty set of destination vertices.…”

Section: A the Thinnest Path Problemmentioning

confidence: 99%

The Thinnest Path Problem

Gao

Zhao

Swami

2013

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Abstract-We formulate and study the thinnest path problem for secure communication in wireless ad hoc networks. The objective is to find a path from a source to its destination that results in the minimum number of nodes overhearing the message by a judicious choice of relaying nodes and their corresponding transmission powers. We adopt a directed hypergraph model of the problem and establish the NP-completeness of the problem in 2-D networks. We then develop two polynomial-time approximation algorithms that offer n 2 and n 2 √ n−1 approximation ratios for general directed hypergraphs (which can model non-isotropic signal propagation in space) and constant approximation ratios for ring hypergraphs (which result from isotropic signal propagation). We also consider the thinnest path problem in 1-D networks and 1-D networks embedded in a 2-D field of eavesdroppers with arbitrary unknown locations (the so-called 1.5-D networks). We propose a linear-complexity algorithm based on nested backward induction that obtains the optimal solution for both 1-D and 1.5-D networks. This algorithm does not require the knowledge of eavesdropper locations and achieves the best performance offered by any algorithm that assumes complete location information of the eavesdroppers.

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Section: A the Thinnest Path Problemmentioning

confidence: 99%

The Thinnest Path Problem

Gao

Zhao

Swami

2013

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“…The cases where the cost is energy consumption [10,11], the number of transmissions [12], or the overhead in route discovery and management [13] have been well studied. Recently, a unified solution to the minimum cost broadcasting problem, where the cost function can have various forms, has been proposed based on the formulation of the so-called neighborhood complex [14]. In contrast to their counterparts in wired networks which have polynomial solutions, the broadcast problems for minimizing the energy consumption and the number of transmissions are shown to be NP-complete in [11,12].…”

Section: A Broadcasting In Sr-sc Networkmentioning

confidence: 99%

“…Furthermore, to our best knowledge, our work is the first to adopt simplicial complexes to model and solve the broadcast problem in wireless ad hoc networks. An attempt to formulating the broadcast problem in wireless ad hoc networks into problems in simplicial complexes can be found in [14]. The simplicial complex is an important topic in algebraic topology.…”

Section: E Related Workmentioning

confidence: 99%

Broadcasting in multi-radio multi-channel wireless networks using simplicial complexes

et al. 2012

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We consider the broadcasting problem in multi-radio multi-channel ad hoc networks. The objective is to minimize the total cost of the network-wide broadcast, where the cost can be of any form that is summable over all the transmissions (e.g., the transmission and reception energy, the price for accessing a specific channel). Our technical approach is based on a simplicial complex model that allows us to capture the broadcast nature of the wireless medium and the heterogeneity across radios and channels. Specifically, we show that broadcasting in multi-radio multi-channel ad hoc networks can be formulated as a minimum spanning problem in simplicial complexes. We establish the NP-completeness of the minimum spanning problem and propose two approximation algorithms with order-optimal performance guarantee. The first approximation algorithm converts the minimum spanning problem in simplical complexes to a minimum connected set cover problem. The second algorithm converts it to a nodeweighted Steiner tree problem under the classic graph model. These two algorithms offer tradeoffs between performance and time-complexity. In a broader context, this work appears to be the first that studies the minimum spanning problem in simplicial complexes and weighted minimum connected set cover problem.

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“…For a more detailed discussion on the potential applications of simplicial complexes in communication and social networks, readers are referred to [12].…”

Section: E Related Workmentioning

confidence: 99%

Broadcasting in Multi-Radio Multi-Channel Wireless Networks using Simplicial Complexes

Ren

Zhao

Ramanathan

et al. 2011

2011 IEEE Eighth International Conference on Mobile Ad-Hoc and Sensor Systems

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We consider the broadcasting problem in multiradio multi-channel ad hoc networks. The objective is to minimize the total broadcast cost, where the cost can be of any form that is summable over all the transmissions (e.g., the transmission and reception energy, the price for accessing a specific channel). Our technical approach is based on a simplicial complex model that allows us to capture the broadcast nature of the wireless medium and the heterogeneity across radios and channels. Specifically, we show that broadcasting in multi-radio multi-channel ad hoc networks can be formulated as a minimum spanning problem in simplicial complexes. We establish the NP-completeness of the minimum spanning problem and propose two approximation algorithms with order-optimal performance guarantee. These two algorithms offer tradeoffs between performance and timecomplexity. In a broader context, this work appears to be the first that studies the minimum spanning problem in simplicial complexes and weighted minimum connected set cover problem.

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Beyond graphs: Capturing groups in networks

Cited by 17 publications

References 21 publications

The Thinnest Path Problem

The Thinnest Path Problem

Broadcasting in multi-radio multi-channel wireless networks using simplicial complexes

Broadcasting in Multi-Radio Multi-Channel Wireless Networks using Simplicial Complexes

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