We consider a large-scale of wireless ad hoc networks whose nodes are distributed randomly in a two-dimensional region Omega (more specifically, a unit square). Given n wireless nodes V, each with transmission range r(n), the wireless networks are often modeled by graph G(V, r(n)) in which two nodes are connected if and only if their Euclidean distance is no more than r(n). We first consider how to relate the transmission range with the number of nodes in a fixed area such that the resulted network can sustain k fault nodes in its neighborhood with high probability when all nodes have the same transmission range. We show that, for a unit-area square region Q, the probability that the network G(V, r(n)) is k-connected is at least e(-e-alpha) when the transmission radius r(n) satisfies npir(n)(2) greater than or equal to In n + (2k - 3)ln In n - 2 ln(k - 1)! + 2alpha for k > 1 and n sufficiently large. This result also applies to mobile networks when the moving of wireless nodes always generates randomly distributed positions. We also conduct extensive simulations to study the practical transmission range to achieve certain probability the network being k-connectivity, when the number of nodes n is not large enough. The relation between the minimum node degree and the connectivity of graph G(V, r) is also studied. Setting the transmission range of all nodes to r(n) guarantees the k-connectivity with high probability, but some nodes may have excessive number of neighbors in the graph G(V, r(n)). We then present a localized method to construct a subgraph of the network topology G(V, r(n)) such that the resulting subgraph is still k-connected but with much fewer communication links maintained. We show that the constructed topology has only 0(k - n) links and is a length spanner. Here a graph H subset of or equal to G is spanner for graph G, if for any two nodes, the length of the shortest path connecting them in H is no more than a small constant factor of the length of the shortest path connecting them in G. Finally, we conduct some simulations to study the practical transmission range to achieve certain probability of k-connected when n is not large enough. Copyright (C) 2004 John Wiley Sons, Ltd.
