MODELING THE SENSOR NODES CONNECTIVITY

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have proposed to study the other extreme and model the sensor network as a general graph; that is, each node can be adjacent to every other node. Model 4.2.2 (General Graph (GG)). The connectivity graph is a general undirected graph G. While a UDG is too optimistic, the GG is often too pessimistic, because the connectivity of most networks is not arbitrary but obeys certain geometric constraints. Still, in some application scenarios it might be accurate to operate either on the UDG or on the GG. Indeed, there are algorithms developed for the UDG which also perform well in more general models. Moreover, some algorithms designed for the GG are currently also the most ef cient ones for UDGs (e.g., reference 2). The research community has searched for connectivity models between the two extremes UDG and GG. For example, the quasi unit disk graph model (QUDG) [3, 4] is a generalization of the UDG that takes imperfections into account as they may arise from non-omnidirectional antennas or small obstacles. These QUDGs are related to so-called civilized graphs. The interested reader can nd more information in reference 5. Model 4.2.3 (Quasi Unit Disk Graph (QUDG)). The nodes are in arbitrary positions in R2 . All pairs of nodes with Euclidean distance at most for some given (0, 1] are adjacent. Pairs with a distance larger than 1 are never in each other s transmission range. Finally, pairs with a distance between and 1 may or may not be neighboring. An example is shown in Figure 4.2. Note that, for = 1, a QUDG is a UDG, and therefore the following theorem holds. Theorem 4.2.1. A UDG is a special case of a QUDG. The QUDG model itself can be extended in several ways.

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Figure 4.2. Quasi unit disk graph from the perspective of node u: Node u is always adjacent to node v1 (d(u, v1 ) ) but never to v5 (d(u, v5 ) > 1). All other nodes may or may not be in u s transmission range. In this example, node u is adjacent to v3 and v4 but not to v2 .

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MODELING SENSOR NETWORKS

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Model 4.2.4 (QUDG Variations). The QUDG as presented in Model 4.2.3 does not specify precisely what happens if the distance is between and 1. There are several options. For example, one could imagine an adversary choosing for each node pair whether they are in each other s transmission range or not. Alternatively, there may be a certain success probability of being adjacent: The corresponding probability distribution could depend on the time and/or distance [6]. For example, the QUDG could be used to study Rayleigh fading; that is, the radio signal intensity could vary according to a Rayleigh distributed random variable. Also, a probabilistic on/off model is reasonable, where in each round a link s state changes from good to bad and vice versa with a given probability. Measurement studies suggest that in an unobstructed environment, and with many nodes available, 1/ is modeled as a small constant [7]. Interestingly, many algorithms can be transferred from the UDG to the QUDG at an additional cost of 1/ 2 [4]. Note that while for .5 this factor is bearable, the algorithms are two orders of magnitude worse if .1. While the QUDG can be attractive to model nodes deployed in elds with few obstacles, it does not make sense for inner-city or in-building networks where obstructions cannot be ignored: Since a node may be able to communicate with another node which is dozens of meters away, but not with a third node being just around the corner, would be close to 0. However, even in such heterogeneous environments, the connectivity graph is still far from being a general graph. Although nodes that are close but on different sides of a wall may not be able to communicate, a node is typically highly connected to the nodes which are in the same room, and thus many neighbors of a node are direct neighbors themselves. In other words, even in regions with many obstacles, the total number of neighbors of a node which are not adjacent is likely to be small. This observation has motivated Model 4.2.6, see reference 8 for more details. Model 4.2.5 (Bounded Independence Graph (BIG)). Let r (u) denote the set of independent nodes that are at most r hops away from node u (i.e., nodes of u s r-neighborhood) in the connectivity graph G. Thus, a set S V of nodes is called independent if all nodes in the set are pairwise not adjacent; that is, for all u, v S, it holds that {u, v} E. Graph G has bounded independence if and only if for all / nodes u G, | r (u)| = O(poly(r)) (typically | r (u)| O(r c ) for a small constant c 2). The BIG model re ects reality quite well and is appropriate in many situations. Figure 4.3 shows a sample scenario with a wall; in contrast to UDG and QUDG, the BIG model captures this situation well. Since the number of independent neighbors in a disk of radius r of a UDG is at most O(r2 ), we have the following fact. The proof is simple (and similar to the upcoming proof of Theorem 4.2.13) and left to the reader as an exercise. Theorem 4.2.2. The UDG model is a special case of the BIG model. Similarly, if is constant, also a QUDG is a BIG.

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