# Position location techniques and applications - Part 2: Ad hoc and sensor network scenarios

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A widely studied problem in the field of MANETs is the connectivity issue. A fundamental property of networks is that every node should be able to communicate with every other node in establishing connections, defining in this way a certain level of connectivity of the nodes, and at the same time the topology of the network.

In a given network, such as the one depicted in Figure 5.9a, mobile node *i* may lose connectivity with the rest of the network simply because it has wandered off too far as in Figure 5.9b, or its power reserve has dropped below a critical threshold, or because of the influence of certain phenomena in the radio channel such as fading or shadowing. In these cases, coverage area is reduced and links with the other nodes of the network are likely affected, as in Figure 5.9c.

**FIGURE 5.9**Connectivity losses of node i by separation and reduction of coverage.

On the other hand, a mobile node in a network, such as node *i* shown in Figure 5.10a, can gain connectivity if it approaches close enough to the rest of the network as in Figure 5.10b, or because it increases its coverage area (as a result of a reload of its power reserve, or by the influence of constructive multipath effects), creating new links with the other nodes, as shown in Figure 5.10c [10].

**FIGURE 5.10**Connectivity gains of node i by approach and increment of coverage area.

In both cases,the connectivity of the nodes and therefore the topology of the network are affected. The analytical techniques contemplated for constructing the theoretical framework include the following methods:

Branching processes.Each individual in the network has a random number of children in the next generation in accordance with a certain probability distribution [38].

Clustering and migration processes.These tell us about the formation of clusters in which individualsmove between groups and can be studied by Markov processes [40, 43].

Random graphs.A graph consists of a set of vertices and edges; a graph is connected if and only if there exists a walk between any two vertices [28].

Point processes.These are stochastic processes where points are scattered over ann-dimensional space in some random manner [54].

Percolation theory.In this formulation, disks with equal radii are generated on an infinite plane according to a Poisson process; as the intensity of the process increases, disks overlap and form clusters. The theorem states that a finite critical intensity exists, above which a unique unbounded cluster almost certainly exists. A cluster of disks is equivalent to a connected network when the maximum transmission range is twice the disk radius. Percolation shows the farthest distance at which the broadcast of packets reaches nodes in the network, providing some long-distance multihop communication. We do not choose this method, because a received packet is not necessarily propagated since it has a specific destination address where routing techniques mark the path and the destination node [49].

Diffusions.A one-dimensional diffusion is a model of the motion of a particle with limited lifetime, continuous path, and no memory, traveling in a linear interval [36, 39].

All these options are worth pursuing as individual research entities. We selected the point processes method. Results and models for connectivity have been developed by Antonio [84].