Fault tolerant deployment and topology control in wireless networks

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Abstract
This paper investigate fault tolerance for wireless ad hoc networks. We consider a large-scale of wireless networks whose nodes are distributed randomly in a unit-area square region. Given n wireless nodes V, each with transmission range rn, the wireless networks are often modeled by graph G(V,rn) in which two nodes are connected if their Euclidean distance is no more than rn.We first consider how the transmission range is related with the number of nodes in a fixed area such that the resulted network can sustain k fault nodes with high probability. We show that, for a unit-area square region, the probability that the network G(V,rn) is (k+1)-connected is at least e-e-α when the transmission radius rn satisfies n π rn2 ≥ ln n + (2k-1) ln ln n -2ln k! + 2α for k0 and n sufficiently large. This result also applies to mobile networks when the moving of wireless nodes always generates randomly distributed positions. Our simulations show that n should be larger than 500 if k=2 or 3 and α = log n and n should be larger than 2500 if k=2 or 3 and α = log log n.We then present a localized method to control the network topology given a (k+1)-faults tolerant deployment G(V,rn) of wireless nodes such that the resulting topology is still (k+1)-faults tolerant but with O(kn) communication links maintained. We show that the constructed topology is also a length spanner. Here a subgraph H is spanner of 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.

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