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On the Phase Transition Width of K-Connectivity in Wireless Multihop Networks
July 2009 (vol. 8 no. 7)
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.

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Index Terms:
Phase transition width, k-connectivity, connectivity, wireless multihop networks, transmission range, average node degree, random geometric graph.
Citation:
Xiaoyuan Ta, Guoqiang Mao, Brian D.O. Anderson, "On the Phase Transition Width of K-Connectivity in Wireless Multihop Networks," IEEE Transactions on Mobile Computing, vol. 8, no. 7, pp. 936-949, July 2009, doi:10.1109/TMC.2008.170
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