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35. F. Guidec and H. Roussain. Asynchronous document dissemination in dynamic ad hoc networks. In Second International Symposium on Parallel and Distributed Processing and Applications (ISPA), 2004. IEEE CD-ROM. 36. A. Lindgren, A. Doria, and O. Schelen. Probabilistic routing in intermittently connected networks. In Proceedings of the Fourth ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2003), June 2003. IEEE CD-ROM. 37. M. Musolesi and C. Mascolo. Controlled epidemic-style dissemination middleware for mobile ad hoc networks. In Mobiquitous, 2006. IEEE CD-ROM. 38. S. Staniford, V. Paxson, and N. Weaver. How to own the Internet in your spare time. Usenix Security Symposium, 2002. 39. C. Shannon and D. Moore. The spread of the witty worm. IEEE Security and Privacy, 2(4): 46 50, 2004. 40. J. W. Mickens and B. D. Noble. Modeling epidemic spreading in mobile environments. In WiSE, 2005. 41. R. Wong and I. Yap. Security information. In Virus Encyclopedia: WINCE BRADOR.A, Technical Details, 2004. Trend Micro Incorporated. 42. P. Ferrie, P. Szor, R. Stanev, and R. Mouritzen. Security Response: SymbOS.Cabir, 2004. Symantec Corporation. 43. S. A. Khayam and H. Radha. A Topologically-aware worm propagation model for wireless sensor networks. In IEEE ICDCS International Workshop on Security in Distributed Computing Systems SDCS, 2005. IEEE CD-ROM. 44. P. De, Y. Liu, and S. K. Das, Modeling node compromise spread in sensor networks using epidemic theory. In World of Wireless, Mobile and Multimedia Networks, WoWMoM, 2006. IEEE CD-ROM. 45. H. Chan, V. D. Gligor, A. Perrig, G. Muralidharan. On the distribution and revocation of cryptographic keys in sensor networks. In IEEE Transactions on Dependable and Secure Computing 2005. IEEE CD-ROM.
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4.1 INTRODUCTION AND MOTIVATION In order to develop algorithms for sensor networks and in order to give mathematical correctness and performance proofs, models for various aspects of sensor networks are needed. This chapter presents and discusses currently used models for sensor networks. Generally, nding good models is a challenging task. On the one hand, a model should be as simple as possible such that the analysis of a given algorithm remains tractable. On the other hand, however, a model must not be too simplistic in the sense that it neglects important properties of the network. A great algorithm in theory may be inef cient or even incorrect in practice if the analysis is based on idealistic assumptions. For example, an algorithm that ignores interference may fail in practice since communication happens over a shared medium. Many models for sensor network have their origin in classic areas of theoretical computer science and applied mathematics. Since the topology of a sensor network can be regarded as a graph, the distributed algorithms community uses models from graph theory, representing nodes by vertices and wireless links by edges. Another crucial ingredient of sensor network models is geometry. Geometry comes into play as the distribution of sensor nodes in space, as well as the propagation range of wireless links, usually adheres to geometric constraints. The chapter is organized as follows. In Section 4.2, the reader will become familiar with various models for the network s connectivity. Connectivity models answer the question: Which nodes are connected to which other nodes and can therefore directly communicate with each other. Section 4.3 then enhances these connectivity models by adding interference aspects: Since sensor nodes communicate over a shared, wireless medium, a transmission may disturb a nearby concurrent transmission. After having studied connectivity and interference issues, we look at
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Algorithms and Protocols for Wireless Sensor Networks, Edited by Azzedine Boukerche Copyright 2009 by John Wiley & Sons Inc.
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modeling questions related to algorithm design in Section 4.4. The reader is provided with a survey of models that in uence the feasibility and ef ciency of certain operations on sensor networks. We draw some general conclusions in Section 4.5, and we point out interesting areas for future research in Section 4.6. 4.2 MODELING THE SENSOR NODES CONNECTIVITY A rst and foremost modeling question concerns the connectivity of sensor nodes: Given a set of nodes distributed in space, we need to specify which nodes can receive a transmission of a node. Throughout this chapter, if a node u is within a node v s transmission range, we say that u is adjacent to v, or, equivalently, that u is a neighbor of v. In the absence of interference (cf. Section 4.3), this relation is typically symmetric (or undirected); that is, if a node u can hear a node v, also v can hear u. The connectivity of a sensor network is described by a graph G = (V, E), where V (vertices) is the set of sensor nodes, and E (edges) describes the adjacency relation between nodes. That is, for two nodes u, v V , (u, v) E if v is adjacent to u. In an undirected graph, it holds that if (u, v) E, then also (v, u) E; that is, edges can be represented by sets {u, v} E rather than tuples. The classic connectivity model is the so-called unit disk graph (UDG) [1]. The name unit disk graph stems from the area of computational geometry; it is a special case of the so-called intersection graph. In this model, nodes having omnidirectional radio antennas that is, antennas with constant gain in all directions are assumed to be deployed in a planar, unobstructed environment. Two nodes are adjacent if and only if they are within each other s transmission range (which is normalized to 1). Model 4.2.1 (Unit Disk Graph (UDG)). Let V R2 be a set of nodes in the twodimensional Euclidean plane. The Euclidean graph G = (V, E) is called unit disk graph if any two nodes are adjacent if and only if their Euclidean distance is at most 1. That is, for arbitrary u, v V , it holds that {u, v} E |u, v| 1. Figure 4.1 depicts an example of a UDG. The UDG model is idealistic: In reality, radios are not omnidirectional, and even small obstacles such as plants can change connectivity. Therefore, some researchers
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Figure 4.1. Unit disk graph: Node u is adjacent to node v (distance 1) but not to node w (distance > 1).
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