Robust wireless ad hoc networks

We consider a large-scale of wireless ad hoc networks whose nodes are distributed randomly in a two-dimensional region /spl Omega/. Given n wireless nodes V, each with transmission range r/sub n/, the wireless networks are often modeled by graph G(V, r/sub n/) in which two nodes are connected if their Euclidean distance is no more than r/sub n/. We show that, for a unit-area square region /spl Omega/, the probability G(V, r/sub n/) being k-connected is at least (e/sup -e/)/sup -/spl sigma// when n/spl pi/(r/sup 2/)/sub n/ /spl ges/ ln n + (2k - 3) ln ln n - 2 ln (k - 1)! + 2/spl sigma/ for k > 1 and n sufficiently large. This result also applies to mobile networks when the moving of wireless nodes always generates randomly and uniformly distributed positions. We also conduct extensive simulations to study the practical transmission range to achieve certain probability of k-connectivity when n is not large enough. The relation between the minimum node degree and the connectivity of graph G(V, r) is also studied.