Threshold behavior of multi-path random key pre-distribution for sparse wireless sensor networks

Wireless sensor network (WSN) has been an active research topic because its application encompasses various domains. In particular, a lot of attention have been paid to the common feature of WSN to show that every node in a large enough network contains certain properties. Real-world applications of random key pre-distribution naturally involve geometric and combinatorial techniques and are even more challenging technically. This paper presents an efficient scheme, which can approximate a complex network by a much simpler object such that the approximation is "regular" between most pairs of partition of this network. Once a more traceable network is obtained, bounds for the probability of the property that a random key pre-distribution subgraph satisfies that each node has a path of length ℓ to its ℓth-hop neighbors are established. Then, by using the sparse version of Szemerédi's regularity lemma and letting C be a constant, n the number of vertices, p the probability of any two nodes sharing at least one common key, a sharp threshold p≥Cn^-(ℓ-1)/ℓ that satisfies this property is shown. Moreover, computer simulations are also given to show the performance of the proposed scheme.