Analysis of Fundamental Limits for Partial Connectivity in Wireless Networks

We consider large, random network topologies, typical in sensor or ad hoc networks. Achieving full connectivity in these networks is hard, as it is known to asymptotically require unbounded node degrees. This certainly involves a lack of scalability, since nodes with finite resources cannot handle an unbounded neighborhood with bounded delay. It does not exclude, however, partial connectivity, which is satisfactory in many applications, such as sensor networks. Therefore, an important step in analyzing the scalability of such networks is to quantify how large part of the random topology can be still expected to belong to a connected component if the nodes are confined to some bounded degree. We investigate this issue in a very general model that contains many of the often used models as special cases. We prove a bound on the asymptotic fraction of nodes that can belong to a connected component. The bound depends on the expected node degrees and is asymptotically sharp.