## Abstract

In this paper, we study the connectivity of wireless ad hoc networks that are composed of unreliable nodes and links by investigating the distribution of the number of isolated nodes. We assume that a wireless ad hoc network consists of n nodes distributed independently and uniformly in a unit-area disk or square. All nodes have the same maximum transmission radius r_{n}, and two nodes have a link if their distance is at most r_{n}. Nodes are active independently with probability 0 < p_{1} ≤ 1, and links are up independently with probability 0 < p_{2} ≤ 1. Nodes are said isolated if they do not have any links to active nodes. We show that if r_{n}= ln n+\xi π p_{1}p_{2}n for some constant ζ, then the total number of isolated nodes (or isolated active nodes, respectively) is asymptotically Poisson with mean e^{-ζ} (or p1e^{-ζ}, respectively). In addition, in the secure wireless networks that adopt m-composite key predistribution schemes, a node is said isolated if it does not have a secure link. Let p denote the probability of the event that two neighbor nodes have a secure link. If all nodes have the same maximum transmission radius r_{n}=ln n+xi π pn, the total number of isolated nodes is asymptotically Poisson with mean e-ζ.

Original language | English |
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Pages (from-to) | 107-124 |

Number of pages | 18 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2010 |

## Keywords

- Asymptotic distribution
- connectivity
- isolated nodes
- random geometric graphs
- random key pre-distribution