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de ne the network topology by forming the proper neighbor relation under certain criteria This is in contrast to the traditional network, in which each node transmits with its maximal transmission power and the topology is built implicitly by routing protocols (that update their routing caches as in timely a way as possible) without considering the power issue A desirable network topology not only reduces energy consumption and prolong network lifetime, but also improves spatial reuse (and hence the network capacity [1]) and mitigate the medium-access control (MAC) level contention The research in topology construction and connectivity has been approached independently along two paths In one path, researchers aim to determine critical conditions on network parameters (such as the transmission range [2 5], the number of neighbors [5,6], the minimum total power required [7 11], or the node failure probability) to ensure network (k-)connectivity with high probability.

Of particular interest is how these critical conditions scale as the number of wireless devices increases In the other path, researchers aim to devise distributed algorithms to enable each node to choose its own transmission power in order to minimize the total transmission power of all wireless nodes, while maintaining (k-)connectivity This problem is, in general, NP-hard in the Euclidean plane [8] What makes the problem more complicated is that there is, in general, no central authority in a multiple-hop wireless network, and each node has to make its decision based on the information collected from the network This implies the topology construction algorithm should be distributed To be less susceptible to mobility, the algorithm should depend only on the information collected locally Algorithms that depend only on local information also incur less message overhead/delay in the process of information collection, and are hence more scalable.

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Several researchers have leveraged elegant graph-theoretic structures to develop localized heuristics [12 19] or ef cient algorithms with bounded approximation ratios [20,21] In this chapter, we give an overview of research activities along these two research thrusts, and present several problems (and their corresponding solutions) related to topology construction, maintenance, and connectivity in wireless sensor networks We consider the problem that arises in topology construction (1) when the network is heterogeneous and nodes may have different attributes (such as transmission radii); and (2) when fault tolerance has to be taken into account (in addition to the primary objectives of network connectivity and reduction in power consumption) The rest of this chapter is organized as follows In Section 102, we introduce the communication models for wireless sensor networks, including the unit-graph model and other probabilistic models In Section 103, we discuss the methods for generating random unit graphs.

It is easy to guess the answer. If the rst bit is 0, we let X = a. If the rst two bits are 10, we let X = b. If we see 11, we let X = c. It is clear that X has the desired distribution. We calculate the average number of fair bits required for generating the random variable, in this case as 1 (1) + 1 (2) + 1 (2) = 1.5 bits. This 2 4 4 is also the entropy of the distribution. Is this unusual No, as the results of this section indicate. The general problem can now be formulated as follows. We are given a sequence of fair coin tosses Z1 , Z2 , . . . , and we wish to generate a discrete random variable X X = {1, 2, . . . , m} with probability mass function

The issue of neighbor discovery and maintenance is treated in Section 104 Following that, we summarize in Section 105 several localized geometric structures for topology construction, and discuss in Section 106 the problem that arises in the case of network heterogeneity Several essential connectivity properties of wireless sensor networks (with the random unit graph as the underlying model) are presented in Section 107 Finally, we discuss the issue of enhancing fault tolerance in topology construction and connectivity in Section 108..

including loss of self-control and risk taking As noted previously, however, these studies are limited due to their correlational design and their exclusive focus on marriage In addition, married people are often mentally and physically healthier than single, divorced or widowed individuals The correlation between marital status and health may have several causes First, it is possible that spouses provide practical support for health behaviors, such as by reminding their partners to keep physicians appointments, eat well, and exercise regularly The social interaction of a marital relationship may also directly increase mental health, which may increase physical health in turn Third, and most relevant here, not being involved in a close relationship may encourage risky, self-defeating behaviors Just as single and divorced people are more likely to take risks while driving, they may also take more risks with their health.

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