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Issue No. 07 - July (2009 vol. 8)
ISSN: 1536-1233
pp: 936-949
Xiaoyuan Ta , The University of Sydney and National ICT Australia, Australia
Guoqiang Mao , The University of Sydney and National ICT Australia, Australia
Brian D.O. Anderson , The Australian National University, Canberra and National ICT Australia, Australia
In this paper, we study the phase transition behavior of k-connectivity (k=1,2,\ldots) in wireless multihop networks where a total of n nodes are randomly and independently distributed following a uniform distribution in the unit cube [0,1]^{d} (d=1,2,3), and each node has a uniform transmission range r(n). It has been shown that the phase transition of k-connectivity becomes sharper as the total number of nodes n increases. In this paper, we investigate how fast such phase transition happens and derive a generic analytical formula for the phase transition width of k-connectivity for large enough n and for any fixed positive integer k in d-dimensional space by resorting to a Poisson approximation for the node placement. This result also applies to mobile networks where nodes always move randomly and independently. Our simulations show that to achieve a good accuracy, n should be larger than 200 when k=1 and d=1; and n should be larger than 600 when k\le 3 and d=2,\ 3. The results in this paper are important for understanding the phase transition phenomenon; and it also provides valuable insight into the design of wireless multihop networks and the understanding of its characteristics.
Phase transition width, k-connectivity, connectivity, wireless multihop networks, transmission range, average node degree, random geometric graph.

X. Ta, G. Mao and B. D. Anderson, "On the Phase Transition Width of K-Connectivity in Wireless Multihop Networks," in IEEE Transactions on Mobile Computing, vol. 8, no. , pp. 936-949, 2008.
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