On the coverage process of random walk in wireless ad hoc and sensor networks

Random walk (RW) is simple to implement and has a better termination control. The Markov chain analysis informs that RW eventually visits all vertices of a connected graph. Due to such nice properties, RW is often proposed for information dissemination or collection from all or part of a large scale unstructured network. The random walker, which can be used to disseminate or collect information, visits the nodes while selecting randomly one of the neighbors. The selection of neighbors is effected by the neighbor density or the connectivity degree of the nodes. The connectivity degree in turn depends on the radius of transmission of wireless nodes. In this paper we studied the coverage process of the RW on random geometric graph. The random geometric graphs are often considered as a model for wireless ad hoc and sensor networks. We defined and studied a metric called "attenuation" that indicates how fast a RW can move in the network while disseminating or collecting information. We showed that attenuation depends on the topology, the number of nodes in a network and the transmission radius of the nodes. We then studied the effect of attenuation on the RW coverage process analytically and through simulations and showed that attenuation is the normalized estimated search time of the network. In the end we applied the results obtained to show that the estimated search time in random geometric graphs is proportional to the reciprocal of the number of replicated targets.