Phase transition [39] | n nodes located randomly in some service area, each assumed to transmit with a fixed radio power in an idealized environment where it can be heard by other nodes within same radius. | Discusses the theory of Bernoulli random graphs [65] that can help in proving when the phase transition in ad hoc network occurs. | Classical theory of random graph models is not suitable to model the ad hoc network problems. |

Minimum node degree and connectivity of wireless multi hop network [40] | 1. Random uniform distribution of nodes and a simple link model 2. All nodes are free to move in the system area according to a certain mobility model 3. Radio link model is assumed | Finds the uniform distribution of homogeneous nodes in a rectangular deployment area and establishes a relationship between the minimum transmission range and the probabilistic behavior of minimum node degree | Results are of theoretical interest and require a very high node density to ensure k-connectivity which would lead to interference and low throughput in real networks |

Minimum node degree and k-connectivity of a wireless multi hop network in bounded area [45] | Circumvent the border effect which assumes boundless network deployment area. | Eliminate the border effects in order to provide an improved estimation of probabilistic characteristic, including the upper bound and lower bound of minimum node degree and k-connectivity. | High node density would lead to interference and low throughput in real networks |

Torus Convention [46] | Nodes are distributed on a unit square according to a homogeneous Poisson point process with density λ. | Eliminate the need to consider boundary effects that may affect the critical transmission range for k-connectivity | It is of theoretical interest, requiring a very high node density to ensure k-connectivity which would lead to interference and low throughput in real networks. |

Critical transmitting range for connectivity in sparse wireless ad hoc networks [47] | Nnodes, each capable of communicating with nodes within a radius of r, randomly and uniformly distributed in d?dimensional region of side of length l. Consider both stationary and mobile nodes | Estimates tolerating parseness i.e. Requiring 90% of nodes to be in the same connected component results in the significant reduction in the required transmission ranges of nodes. | If physical node degree is upper bounded by a constant, then the resulting communication graph is disconnected |

Connectivity in finite ad hoc networks [48] | Uniform distribution of nodes in [0,z] where z > 0 for one dimensional network | Finds the probability of connectivity of one dimensional finite ad hoc network formed by uniform distribution of nodes | Probability of network was not correct and has been corrected by [68] |

Cell extension and Mobility patterns [49] | All relay nodes are randomly distributed and fixed. Mobile nodes and relay nodes move right at same velocity. Relay nodes are classified into two groups. In first group both relay nodes never move and in other group relay nodes move left at velocity | Finds that multi-hop networking with two kind of mobile relay nodes degrades cell extension performance compared with fixed relay nodes. | It is of theoretical interest and cannot be apply to real scenarios. Also, Distributions of mobile nodes are considered only on street not on plane. |

Asymptotic critical transmission radius and critical neighbor number for k-connectivity [50] | Uniform n point process over a unit area disk or square. | Obtained the improved asymptotic almost sure upper bound on the critical neighbor number for k-connectivity. | It does not consider interference, though dense networks produce strong interference. Decrease the network throughput. |