Connectivity is one of the fundamental and widely studied properties of wireless multihop ad-hoc networks. The existing studies applied the theories in the following area:
1. Random Geographic Graph: E.g. for uniformly distributed nodes, the longest nearest-neighbour link and the longest MST (minimum spanning tree) edge have the same value as n goes to infinity. -> connectivity occurs (asymptotically) when the last isolated node disappears.
2. Continuum Percolation: Refer to [2] and [3].
However, as discussed in [1], the theories are only applicable to dense networks. The contribution of the results on dense networks may be limited as network capacity will be compromised. In order to circumvent this problem, the authors added the size of the deployment region as a parameter of the model, and characterise the critical transmission range as the size goes to infinity.
The following two theorems are obtained in [1] for 2D networks:
[1] P. Santi and D. M. Blough, "The Critical Transmitting Range for Connectivity in Sparse Wireless Ad Hoc Networks," IEEE Transactions on Mobile Computing, vol. 2, pp. 25-39, 2003.
[2] P. Gupta and P. R. Kumar, "Critical power for asymptotic connectivity in wireless networks," Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang (Eds.), pp. 547-566, 1998.
[3] O. Dousse, et al., "Connectivity in ad-hoc and hybrid networks," in IEEE Conference on Computer Communications (INFOCOM), 2002, pp. 1079-1088.
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